
Copyright N? 



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COPYRIGHT DEPOSnV 



NOTES 






MECHANICS 



BY 



CHARLES R. CROSS 



Printed for the use of Students 



IN THE 



Massachusetts Institute of Technology 



1908 



BOSTON 

WM. B. LIBBY, THE GARDEN PRESS 

16 ARLINGTON STREET 



jUfiMAftY or OoNtaliCSS] 



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Copyright, 1908 
By CHARLES R. CROSS 



MECHANICS. 



DEFINITIONS. 

1. Motion. — Motion is change of position in space. We can be 
acquainted with none but relative motions, as we can know that a body 
really changes its position only by comparison with some other body not 
possessing the same movement. Our use of the term " rest " is also relative. 

In pure motion of translation all the points of a body move with the 
same velocity and in the same direction at the same instant. When the 
points of a body describe arcs of concentric circles about some fixed axis, 
the motion is one of pure rotation. All possible varieties of motion 
may be produced by the combination of translation and rotation. 

2. Velocity. — As the term is commonly used, velocity is speed, or 

ds 
rate of motion. It is expressed analytically by the formula v — — — 

at 

For uniform motion it is evident that speed equals the space traversed in 

any given time divided by that time. 

In physical treatises, however, it is usual to distinguish between velocity 
and speed, making the former a vector quantity, involving both magni- 
tude and direction, while the latter denotes magnitude only, as defined 
above. 

It is furthermore evident that the speed of a particle at any instant is 
always the space which it would describe in a unit of time were its speed 
to remain of the same value during that time. 

It is customary to express speed in terms of the number of units of length 
which would be thus traversed in a second; e.g., a freely falling body at 
the end of its first second of fall has a speed of 9.8 meters per second. 

3. Force. — We have come to ascribe every change that occurs in the 
condition or qualities of matter to the action of force. Force is commonly 
defined as that which causes or tends to cause a change of condition. A 
mechanical force, or pondero-motive force as it is sometimes called, tends 
to produce motion of a mass. 

An electro-motive force, on the other hand, tends to produce a flow of 
electricity. A magneto-motive force tends to produce magnetic flux. 
But these are not forces in a mechanical sense. 

It will readily be seen that what we call "force" is really known to us 
only as a conception. All that we learn directly from experience is that 
given certain conditions certain results invariably follow. We conceive 
of something which we call force as acting to bring about these results. 



2 MECHANICS. 

Thus when a stone is raised above the surface of the earth it tends to 
fall to the earth, and will do so if unsupported, while if supported it exerts 
a pressure upon the support. This tendency we ascribe to a force which 
we call ''gravity" acting between the stone and the earth. But all that 
we know experimentally is that the stone and earth tend to approach each 
other, and that this tendency has a definite magnitude in any particular 
case. 

Also, when we speak of a force as acting for a certain time, the only 
fact that we know from observation is that the conditions under which 
certain actions take place persist for a certain time. 

4. Composition of Velocities and Forces. — We shall for the 
present consider only the case in which these are applied at a point. 

Velocities and forces are vector quantities. Hence the combined effect 
or resultant of a number of velocities or forces can be found by the usual 
process of vector addition. This is the case whether the components lie 
in the same or in different planes. 

Applications of this principle are the familiar propositions known as 
the Parallelogram and Triangle of Velocities, the Parallelogram and 
Triangle of Forces, the Polygon of Forces, the Parallelopiped of Forces. 

5. Equilibrium of Forces.— It follows from the facts stated in the 
preceding paragraph that two forces are in equihbrium when they are 
equal and opposite, since their resultant is then equal to zero. Also for 
a like reason any number of forces are in equilibrium when they can be 
represented by all the sides of a polygon taken in order. 

6. Couple. — A moment {torque) or a couple (see § 29, p. 9) is also 
a vector quantity, since it acts to produce pure rotation in a definite plane. 
It is customary to represent a moment {jorce X arm) by a distance laid 
off on the axis, right-handed rotations being made positive. Couples may 
be combined by ordinary vector addition, as will be explained later. 

7. Pressure and Impulse. — An impulse is to be considered merely 
as a pressure acting for a very short time. Any pressure may be considered 
as due to a series of recurring impulses, separated by very short intervals. 
Such, for example, is the explanation of gaseous pressure offered by the 
Kinetic Theory. 

8. Mass. — The term " mass" is commonly used to denote the quantity 
of matter that a body contains. 

9. Measure of Forces. — We shall shortly see that all mechanical 
forces may be compared by their effects in producing or modifying motion. 

Steady forces may be measured statically by the use of various forms of 

dynamometer, of which the ordinary spring balance is an example. 

ID. Acceleration. — Acceleration is rate of change of velocity. It is 

dv d^s 
expressed by the equation a = -— = -— • 



THREE LAWS OF MOTION. 3 

It is evident that if the acceleration is constant, — that is, if equal incre- 
ments of velocity are gained in successive equal times, the acceleration is 
equal to the velocity gained in a unit of time. 

Thus the acceleration of a body falKng freely under the influence of its 
weight is 9.8 meters per second per second; often expressed as 9.8 meter / 
(second)^. As the unit of time habitually used in physical measurements 
is the second, this may be more briefly stated as 9.8 meters, the expression 
"per second per second" being understood. 

11. Mechanics. — Mechanics, in the sense in which the term is ordi- 
narily used, is that branch of physics which treats of the action of force on 
bodies. It is commonly divided into Statics and D}Tiamics. Statics 
treats of balanced forces, or forces in equihbrium; Dynamics, of the action 
of forces in producing motion. A better nomenclature, however, is that 
used by Thomson and Tait, as shown in the following quotation from their 
"Elements of Natural Philosophy" (1873): — 

" The science which investigates the action of force is called by the most 
logical WTiters Dynamics. It is commonly but erroneously called Me- 
chanics; a term employed by Ne\\ton in its true sense, the Science of 
Machines and the Art of making them. 

"Eorce is recognized as acting in two ways: (i) so as to compel rest 
or to prevent change of motion; and (2) so as to produce or to change 
motion. D^Tiamics, therefore, is divided into two parts which are con- 
veniently called Statics and Kinetics." 

THREE LAWS OF MOTION. 

The following propositions, first clearly and collectively stated by New- 
ton, are shown to be true by universal experience. The paragraphs in 
quotation marks are given in translation as originally stated by Newton 
in the "Principia," pubKshed in 1687. 

12. Law I. — '^ Every body conti?mes in its state of rest, or of uniform 
motion in a straight line, except in so far as it may he compelled by force 
to change that stated 

Another mode of stating the law is the following: A body at rest con- 
tinues in that state, and a body in motion proceeds uniformly in a straight 
line, unless acted upo?i by some external force. This truth is the principle 
of the Inertia of Matter. 

Illustrations of Law L: Railroad accidents; phenomena observed 
by standing passenger on Boston and other street railway cars; coursing; 
fixing head of hammer. 

13. Law II. — ^^ Change of motion is proportional to the force applied, 
and takes place in the direction in which that force acts.^' The term '' change 



4 MECHANICS. 

of motion," as used by Newton, is identical with the term "change of 
momentum," now universally employed. 

14. Momentum. — The product MF of a mass M by its velocity V is 
called its momentum, which is evidently proportional to both M and V. 
If we call a the acceleration produced by any force F acting upon a mass 
M, the rate of change of momentum will be the mass multiplied by the 
rate of change of its velocity, or Ma. Hence the following statement is 
true: Forces are proportional to the momenta which would be generated 
by their constant and uniform action during a unit of time, or equal times; 
or, in other words, the unbalanced force acting upon a body is measured by 
the rate of change of momentum thereby produced. Algebraically this is 
indicated by the expression F a Ma, in which F is the force producing 
an acceleration a in the mass M. 

If the force F acts in opposition to a motion already existing in the 
body, a may be regarded as negative. 

If we suppose the same force F to act on different masses Mi, M^, pro- 
ducing accelerations a^, a^, respectively, we shall have F oc JV/j a^, F oc 
^2^2, wheruce M^a^^ = M^Oz and ^i : aj : : M2 : Mj. Hence, The velocities 
impressed upon different masses by the action of equal forces during a unit 
of time, or during equal times, are inversely proportional to those masses. 

The following propositions are derived immediately from the relation 
F a Ma. 

If M is constant, F is proportional to a; whence. Forces are propor- 
tional to the accelerations which they impress upon equal masses. 

If a is constant, F varies as M, whence, Forces are proportional to the 
masses upon which they impress equal accelerations. 

It will be shown later that the momentum MV generated by the action 
of a force i^ on a mass M for a time / is equal to the magnitude of the 
force multiplied by its duration; i.e., Ft = MV. The quantity Ft is 
called the impidse of the force. Hence the effect of a force is measured 
by its magnitude multiplied by the time during which it acts. 

15. Corollary to Law II. — When any number of forces act simul- 
taneously upon a body, then, whether the body be originally at rest or in motion, 
each force produces exactly the same effect in magnitude and direction as if 
it acted alone. This principle is often .called the Law of the Independence 
of Motions. 

Illustrations: Man walking or writing on vessel; Parry's sailors on 
ice-floe; body falling from an elevation; cannon-ball fired in different 
directions with regard to motion of earth. 

16. Law III. — " To every action there is always an equal and contrary 
reaction; or, the mutual actions oj two bodies are always equal and oppositely 
directed. ^^ 



ANALYTICAL STATICS. 5 

Illustrations: Attraction or repulsion of magnets; action of uncoiling 
spring; recoil of gun; mutual attraction of earth and mass. 

17. Time Required to Produce Motion of a Mass. — WTien a 
force is applied to a large mass the mass necessarily acquires velocity only 
with comparative slowness; that is, the acceleration is small. The greater 
the mass, with a given force, the more slowly does the mass gain in speed. 
Hence if the force is applied to the mass, for example, by means of a spring 
or cord, the more sudden the application of the force, the more will the 
spring or cord be stretched. The mass seems to offer a resistance to enter- 
ing into motion, though it is nevertheless acquiring velocity at the rate 

. . F 

conditioned bv the 2d Law of Motion; i.e., a == « 

M 

This effect is often said to be due to the " time required to overcome the 
inertia of a body." Such a statement is erroneous if the word "inertia" 
is used in its strict sense; viz., as denoting the absolute inability of matter 
to change its state except under the action of force. But this term is very 
commonly used in a somewhat different though analogous sense as denot- 
ing that property of matter in virtue of which a definite force is necessary 
to produce a given change in the existing state of a mass. This last is 
simply a mode of expressing the general idea of momentum. The same 
idea is often expressed by the statement that " time is required to produce 
motion in a mass." 

Many peculiar phenomena are explained in virtue of this action; as, for 
example, the feat of firing a candle through a board, the breaking of the 
harness under the sudden pull of a horse, the bursting of cannon with 
high-power explosives, surface-blasting with dynamite, and the like. 

The three Laws of Motion, as is the case with all physical principles, 
are known to be true only from observation and experiment. The best 
proof that we have of their universality is found in the accordance of 
observed with predicted results, as for example in Astronomy in the case 
of eclipses, occultations, planetary and cometary motions and other phe- 
nomena in which the mutual actions of masses of matter are concerned. 

Analytical statics. 

18. Analytical Statics. — The following elementar}' appHcations of 
the principles of analytical statics are inserted here for convenience of 
reference. 

19. Composition and Resolution of any 
Number of Forces Applied at a Point, {a) 
Forces in one Plane. — Referring to Fig, i, it will 
be seen that if ^IC represents any force R, then 
AD and AB will represent the rectangular com- 
ponents of this force. Hence calling these com- 
ponents i^i, F2 respectively and denoting by a the 
angle made by F^ with i?, we have ^ig. 

F^ = R sin a, F2 = R cos a. 




6 MECHANICS. 

It will also appear from a consideration of Fig. i that if several 
forces F^, F^, F^, etc., in the same plane and applied at a point, make 
angles a^, a^, a^, with the axis of X, the sum of their components resolved 
parallel to X \s Fx = ^F cos a, and the sum of their components parallel 
to Y \s Fy = IF sin a. Since the resultant R = \/p 2 _|_ p^2 ^^ h'dve 
for its value R = \/{IF sin ay + {IF cos ay, 

y ft siji /y 

Also, if is the angle made by R with A^, tan = -— ^^ 

2p cos a 

For equilibrium IF sin a = o, IF cos a = o. 

(b) Forces in Space. — For a force in space it is easy to see that if the 
point of appHcation be taken as the origin of coordinates, the components 
Fx, Fy, Fz, along the axes are respectively Fx = F cos a, Fy = F cos ^, 
Fz =^F cos )-, a, /?, 7- being the direction angles of the force F. Also, 
F^ = Fx^ + jP/ + Fz^- Hence for a system of forces 

^ = V(IF cos ay + (IF cos ^y + (IF cos rt". 

Calling ar, ^r, fr, the direction angles of R, we have 



cos av 



IFx IF cos a 



cos /?r = 



R R 

IFy _ IF cos /? 

I Fz _ IF cos ;' 



The conditions of equilibrium with such a system are 

IF cos a = o, IF cos /? = o, 2'F cos ^^ = o. 

20. Center of Parallel Forces.— The center of a system of parallel 
forces is the point through which the resultant always passes, however 
the direction of the component forces, while still remaining parallel, may 
be shifted. 

It may be found as follows: — 

Suppose Xr, jr, Zr, to be the coordinates of the center, x, y, z, being those 
of the point of appHcation of any component F. If we assume the com- 
ponent forces to be made parallel to Y, it follows that XrIF = IFx, 
since the moment of the resultant R = IF, relatively to the axis Z, must 
be equal to the sum of the moments of the various components relatively 
to the same axis, i.e., to IFx. In like manner, if the components be 
made parallel to Z and to X, yrIF = IFy, ZrIF = IFz. These equa- 
tions give the values of Xr, yr, Zr, and so determine the position of the center. 

21. Center of Mass, Center of Gravity, Center of Inertia. 
— These terms all apply to the same point, which is the center of the system 
of parallel forces formed by the weights of the particles composing the 
body or system of bodies. 

It follows from the last preceding demonstration that for a system of 
masses the coordinates of the center of mass are 

^ IWx IMx _ IWy _ I My ^ IWz ^ IMz 

^' ~ IW ~ IM ' ^^ ~ ~I\V ~ IM ' ^' ~ IW ~ IM ' 
For a single body of mass M, 

JxdM j'ydM fzdM 

jdM jdM jdM 



PARTICULAR CASES OF RECTILINEAR MOTION. 

If the body is homogeneous, 

fxdV jydV fzdV 



OCr — ^ 1 yr — ^ f Zr — 



JiF pF JrfF 

The integrals must, of course, be taken between the proper Hmits. 
If the origin is at the center of mass, :x:r = o, > = o, Zr = o; whence 
CxdM = o, CydM = o, CzdM = o. 

PARTICULAR CASES OF RECTILINEAR MOTION. 

22. Uniform Motion. — It appears from the foregoing that a body 
may remain at rest either when acted upon by no force whatever, or under 
the action of a system of balanced forces, since in each case the effective 
force acting is o. The second case is the only one actually reahzed. 

Also a body may move uniformly in a straight line, either under the 
action of no force or under the action of a system of balanced forces. In 
the second case, the only one actually realized, the body is often said to be 
in dynamical equihbrium. 

Illustrations of such a condition of motion are found in the uniform motion 
of a steamer when the motive force is just equal to the resistance of the 
water and air; and in the case of a railway train when the motive force is 
just balanced by frictional and other resistances. 

23. Uniformly Variable Motion.— A body acted upon by an un^ 
balanced force moves with a variable motion. The most important case 
is that in which the force is constant and in the direction of the body's 
motion, in which case the acceleration is constant and the motion is uni- 
formly accelerated. 

dv 
That the acceleration, — = a, is constant follows immediately from 

dt 

the 2d Law of Motion, since the effect of a constant force is to produce a 

constant rate of change of speed. 

dv d^s 

It is furthermore true that a = — = Hence by integration be- 

dt dt^ 

tween the limits o and t we reach the famihar formulae for uniformly ac- 
celerated motion when the body starts from a state of rest under the action 
of the accelerating force, viz.: — 

V = at (i) s = i at^ (2) V = \/7a7 (3). 
Equation (3) is obtained by eliminating / from (i) and (2). 



For freely falling bodies we have v = gt, h = \ gt^, v = \/2gh, 
where h is the distance fallen through and g the acceleration due to 
gravity. 



8 



MECHANICS. 



If the body possesses an initial velocity v^ when the accelerating force 
begins its action, vi = v^ + «/, st = vjt + h ^^^y '^'h ^h being respectively 
the velocity acquired and space traversed in / seconds. 

If the force acts in opposition to an already-acquired motion, the accel- 
eration is negative and the motion is uniformly retarded. In this case 

Vl = 7'o — <-^^j ^t = '<-'oi — h^i^' Also Si = 

2 a 

24. Measure of Impulse.— The origin of the equation Ft = MF will 
now be clear. If a force F acts for a time / on a mass M, it is evident that 
the resulting change in velocity is V = at. But F = Ma; whence 
Ft = Mat = MV. That is, the impulse is measured by the resulting 
change of momentum. 

25. Comparison of Forces with Gravity. — If a force F acts on a 
mass of weight IF, the acceleration a imparted can be determined from 

F 

the proportion F -.W : : a : g, whence a = g 

W 

26. Motion over Inclined Plane. — It will be seen, by a reference to 

Fig. 2, that for a body of weight W 
descending a frictionless inclined 
plane, of height H and length L, the 

H 

^ accelerating force will be F = IF 

Hence, as F : W : : a : g, the accele- 

H 

ration of the body will be ^ = g 

As a is constant, the motion will be 
uniformly accelerated. By substitu- 
ting this value of (7 in the general 

formulae for imiformly accelerated motion, we derive formula? for the case 

of a body descending an inclined plane, as follows: — 




H H I H 

L L ^ '^ L 



H 



Denoting the angle of slope of the inclined plane by 0, we have — = sin 



and hence 



V = gt sin 0, s = igt^ sin 0, 



V = \/ 



V 2gs sin 0. 



ADDITIONAL PROPOSITIONS. 



ADDITIONAL PROPOSITIONS. 



27. Resultant Momentum. — The resultant momentum in any direc- 
tion of a system of bodies is Mr = Imv, v being the resolved component of 
the velocity of m in the direction considered. Thus the resultant momentum 
of a system of particles parallel to X is Mx = Ilmvx = 2mv cos a, that 
parallel to F, My = Imvy = I!mv cos j^, that parallel to Z, IMz = 
I mvz = 2niv cos T. Also Mr = \/Mx'' + Af/ +1/2^ 

28. Relation to Center of Mass.— The resultant momentum of 
any system of bodies is the same as if they were concentrated at the center 
of mass of the system. 

Let Wj, W2 be the masses of two bodies at distances Ij, 4 from any assumed 
coordinate plane, and let Ig be the distance of their center of mass from 
the same plane. Then, since the moment, relatively to the plane, of the 
sum of the masses assumed to be concentrated at their center of mass is 
the same as the sum of their separate moments (see § 21, p. 6), we have 
(mi + m^) Ig = mA + ^24- 

Taking the derivative of each term relatively to time we have (Wi + m^ 

dig dL tt/2 -n 1 1 • • , . . r 

-—^ = in, "V" + ^2 -77- -DUt the derivatives represent velocities of w., Wj, 
dt at at 

separately and of (w^ + nis) assumed to be concentrated at their center 
of mass. Calling these Vj, v^, Vg, respectively w^e have (Wj + Wa) Vg = 
fttjV^ + msVz' The second member of the equation represents the resultant 
momentum of Wj, Wj, and the first member represents the momentum of 
the masses concentrated as assumed. 

The proof is general, for with any number of additional masses what- 
soever we may follow the same process, combining them successively. 
Thus we may apply the same reasoning as before to a third mass W3 in 
connection with the masses Wj, Wg supposed to be concentrated at their 
center of mass; and so on. 

It also follows from what precedes that the resultant momentum of any 
system of bodies relatively to their center of mass is zero. 

29. Couple. — If two equal and opposite parallel forces, not acting 
in the same straight line, are applied to a body, their algebraic sum is 
zero, and hence there is no tendency toward motion of translation. But 
as their resultant moment relative to any point can never become zero, 
their sole effect will be to cause rotation about an axis. 

The rotary effect, torque or moment of a couple is measured by the 
product of either force into the length of the 
arm. 

Let Fj F' (Fig. 3) constitute a couple 
whose arm is yl 5. To find the rotary effect, ^_ 
let P be any point. The moments of F, 
F', relatively to P, are F X AP, F' X BP. 
Hence, the total resultant moment of the two F' 

forces '\^F y. AP ^- F' y^BP = FY. AB. ^^^ ^• 



r 



B 



lO MECHANICS. 

If the point be taken on the same side of both forces, as at P\ the resultant 
moment is F' X P'B - F X P'A ^ F X AB, as before. Hence T = Fl, 
where / is the arm. 

A couple may be turned in its own plane or moved parallel to itself 
without altering its efficiency, since the product Fl remains constant. 
Also, any couple may be replaced by an equivalent one having a given 

T 
arm. If l^ be the arm, the corresponding force will be F^ = — • 

30. Combination of Couples having the Same Axis. — The 

resultant moment of any number of couples lying in the same or parallel 
planes is equal to the algebraic sum of their separate moments. 

Calling Tr the resultant moment, Tr = ^Fl. For equilibrium, so 
far as rotation is concerned, IFl = o. 

Couples not having the same axis may be compounded by a process 
similar to that used in compounding oblique forces. (See § 94, p. 40.) 

Evidently a couple cannot be balanced by any single force, but only 
by the application of an equal and opposite couple. 

31. Total Effect of Force on Free Body.— The tendency of any 
force acting upon a body is, in general, to produce (i) a translatory 
motion, and (2) a rotation. If the force acts through the center of mass 
of the body, the resultant moment relatively to that point is zero, and there 
is no tendency to rotation if the body is free. This will be clear if we 
consider the force in question to be resolved into an infinite number of 
parallel forces applied to each particle of the body, and proportional to 
the mass of that particle. These would tend to cause equal accelerations 
in each particle, and hence to cause all points in the body to move with 
the same velocity. 

Because of this property of the center of mass it is often called the center 

oj inertia. 

If the line of action of the force does not pass through the center of 

mass, the total translatory effect will be the same as if the force were ap- 
plied at that point, and the rotary 
effect will be equal to its moment with 
regard to the center of mass. In 
Fig. 4 let F be a force applied at 
A. The condition of the body will 
not be altered if we imagine two op- 
posite forces, F^, F2, each equal and 
parallel to F, to be applied at G, the 
center of gravity. But we have now 

a couple whose moment is T*' X AG, and a force F^ equal and parallel to 

F, and passing through G. 




MEASUREMENT OF FORCE AND MASS, IT 

It can be shown that the rotation produced by the couple F X AG will 
take place about an axis passing through the center of mass of the body. 

Every possible motion of a rigid body may be considered as com- 
pounded of a rotation about its center of mass combined with a translatory 
motion. 

It follows from what precedes that any number of forces acting upon 
a body or system of bodies may be replaced by a single resultant force 
passing through the center of mass, and a resultant couple. 

32. Transferability of Force. — Since the translatory effect of a 
force is solely to produce motion in its Kne of action while the rotary effect 
is its moment, the total mechanical effect of a force will not be altered by 
transferring its point of appHcation to any point in its line of action. 

This principle facilitates the process of composition of two or more 
obhque forces not applied at the same point. The lines of action of any 
two components may be prolonged till they meet in a point, in which case 
the Parallelogram of Forces becomes directly applicable. The remaining 
components can be combined successively in the same manner. 

MEASUREMENT OF FORCE AND MASS. 

33. Units and Standards. — As a prehminary to the discussion of 
this subject it will be necessary to explain briefly the nature of the units of 
length, mass, and time which are adopted in scientific and industrial 
measurements. 

The unit of length in the metric system is the distance between two 
transverse lines ruled on the surface of a certain standard platinum-iridium 
bar (a line standard) adopted as such in 1889 and kept at Paris, which is 
known as the International prototype standard meter. The standard 
temperature at which the bar is correct is 0° C. This length is as exactly 
as possible the length at the same temperature of the original " Metre des 
Archives" of 1799, which is an end standard. For scientific purposes, 
however, the centimeter, one one-hundredth part of the meter, has been 
found to be preferable to the meter for use as a unit. 

The unit of length in the British system is the Imperial yard, which is 
the distance between two transverse lines ruled on gold studs inserted in 
a certain bronze bar (a line standard), when at a temperature of 62° F. 
This bar was made the legal standard of Great Britain in 1855 and is kept 
in the Standards Ofl&ce, Westminster, London. The foot is habitually 
employed as a more convenient practical unit than the yard. 

The legal yard of the United States is defined (1893) as being of such 
length that 39.37 of its inches shall equal the length of the meter. 

The unit of mass in the metric system is the gram. This is a mass equal 



12 MECHANICS. 

to one one-thousandth part of the mass of the International standard pro- 
totype kilogram, a platinum-iridium cylinder, adopted as a standard in 
1889, and placed together with the International prototype meter in the 
care of the International Metric Commission at St. Cloud, Paris. This kilo- 
gram is an exact copy of the original "Kilogramme des Archives" of 1799. 

The British unit of mass is the Imperial avoirdupois pound. This is 
represented by a certain cylindrical piece of platinum, legalized in 1855 
and kept at Westminster. 

The legal avoirdupois pound of the United States is a pound of such 
mass that 2.2046 pounds shall equal one kilogram. 

Copies of the prototype standard meter and kilogram ("national pro- 
totype standards") are in the possession of each of the principal nations. 
Those belonging to the United States are kept at the Office of Weights 
and Measures in Washington. 

The standard meter is found by comparison to be equal to 39.370113 
inches of the British standard yard. The standard kilogram is equal 
to 15432.3564 grains of the British standard pound. This is the legally 
adopted ratio in Great Britain. 

The unit of time universally employed in both the Metric and British sys- 
tems is the ordinary sexagesimal second of civil life, which is the -^ — part 

of the length of the mean solar day. 

It was originally intended that the meter should be one ten-millionth 
part of a quadrant of the meridian in length. The actual meter, however, 
is slightly shorter than this. There is, moreover, apparently a small 
difference in the length of the different meridians according to the longitude. 
The mean length of a quadrant of the meridian is in fact 10,002,000 meters 
instead of 10,000,000 meters. (Clarke, 1880.) 

It was also intended that the gram should be precisely the mass of a 
cubic centimeter of water at the temperature of its maximum density, 
4° Centigrade. The actual mass of a cubic centimeter of water at 4° C. 
is, in fact, 0.999972 grams as determined at the International Metric 
Bureau (1907). 

Inasmuch as there is always a possibility of a slow secular change in 
the molecular constitution of a metal bar and consequently of a minute change 
in its length, the use of the wave-length of a selected kind of homogeneous 
light as a standard has l)een suggested at different times by several persons, 
first by Lamont (1823). Michelson has recently (1892-93) determined 
with great accuracy and bv a direct method the IcMiglh of >uch a wave, 
])r<)duced by incandescent cadmium vapor, in terms of the present ])roto- 
type standard meter, and hence determined the length of tlie meter in 
terms of this wave-lengtli. 

Tile meter contains 1553163.6 wave-lengtlis of a certain com])onenl 
(red) ray of ca(hiiium light. 

liy repealing this measurcnu'nt inanv \('ai> Iumuc il will \)c possibk* to 
ascertain wiietlier the blandard meter ha> in an\- \\a\' altered in length. 



MEASUREMENT OF FORCE AND MASS. 1 3 

Also if the present standard and all accurate copies of it were to be de- 
stroyed it could still be reproduced at any time from this wave-length. 

In view of the possible question as to a sensible variation in the length of 
the day, other units of time than the second have been proposed, as, for 
example, the period of vibration of an elastic spring of determined material 
and dimensions under fixed conditions of temperature, etc., "a perennial 
spring," to use Kelvin's term. A tuning-fork would be the most suitable 
instrument for the purpose. But aside from uncertainty as to exact con- 
stancy of external conditions, the liability to a secular change of rate of 
the fork arising from slow internal molecular changes would make the use 
of such a standard unpractical. 

It was suggested by Maxwell that the period of vibration of a homo- 
geneous light-wave emitted by incandescent vapor, that of sodium, for 
instance, might be used. This period is exact and has been accurately 
determined in terms of the present second. An objection, however, is its 
excessive smallness, approximately 510 X lo"^^ seconds. 

34. Dynamical Measure of Force. — Any mechanical force, whether 
pressure or impulse, may be measured by means of a system based upon 
the 2d Law of Motion. 

By making a suitable choice of units we may write F = Ma instead of 
merely F a Ma. 

The unit of force chosen is that force which produces an acceleration 
unity in a mass unity. This is indicated algebraically in the equation. 
F = Ma, as by making M and a each equal to unity, F also becomes equal 
to unity. 

F 

Since F = Ma, it follows also that M = — , that is, the mass of a body 

a 

may be measured by the constant ratio between any force and the ac- 
celeration which that force will produce when acting on the body. 

Hence, if W is the force by which a given body of mass M is drawn 
toward the earth, that is, if W is the weight of the body, and g the accele- 
ration due to gravity, we have, since g is an acceleration produced by a 

W 
force W acting upon that body, M = — . 

g 

The unit of mass in all systems will weigh (i. e., be drawn to the earth 

W 
by a force of) g units of force. For, since M = — , then, when M equals 

g 

unity, W must be numerically equal to g; that is, the unit of mass must 

be drawn to the earth by a force of g units of force. Likewise, the unit of 

force must be — part of the weight of the unit of mass. 
g 

W 

It is evident that the equation M = — is independent of the particular 

g 
imits of mass, force, and length that we may choose to adopt. 



14 MECHANICS. 

It will be clear from what has been said that either the unit of mass or 
the unit of force may be selected arbitrarily, and the corresponding unit of 
force or of mass determined in accordance with the relations which have 
been shown to exist between them. 

Two principal systems of measurement of force and mass have been 
used, called respectively the Absolute and the Gravitation System. The 
former selects the unit of mass and determines the corresponding unit of 
force; the latter selects the unit of force and determines the corresponding 
unit of mass. 

35. Absolute System. — In this system, the mass of the gram or pound 
is chosen as the unit of mass. The mass of a body is therefore expressed 
either in grams or in pounds. 

I 

The unit of force is determined from this, it being — part of the weight 

g 
of the gram or pound. The value of g is commonly expressed in centimeters 
or in feet, according as metric or British measures are used. 

These units were originally proposed by Gauss. It is evident that 
they are absolutely constant in all places and under all conditions. 

The Absolute System is used in all refined physical measurements. 

We shall see later that units of measurement of all physical forces can 
be derived from the fundamental units of length, mass, and time. Such 
units are called absolute units, or, more logically, derived units. 

The particular form of absolute system now universally employed in 
science is the Centimeter-gram-second System (C. G. S. System), so called 
from the fundamental units of length, mass, and time on which it is based. 

The C. G. S. unit of force is called a dyne, and is — of the weight of a gram, 

g 
g being expressed in centimeters. The British Foot-pound-second System 

is based upon the foot, pound, and second. The corresponding unit of 

I 

force is cdllcd a poundal, Sind is — of the weight of a pound, g being expressed 

g 
in feet. This system, however, is practically obsolete. 

Various other forms of the Absolute System have been used in past 
times; as, for example, a meter-gram-second system, a millimeter- gram- 
second system, and a joot- grain-second system. 

It will appear on consideration that the absolute unit of force thus 
derived will fulfil the requirements of the general definition of a unit of 
force, and generate an acceleration unity in a mass unity. Thus, for ex- 
amj)le, the dyne must generate an acceleration of one centimeter in a mass 
of one gram. For if such a mass falls freely, it will acquire an acceleration 
of g centimeters. But the accelerating force is g dynes, that being the 
weight of a gram. Hence, under the action of a force of one dyne the 
acceleration acquired would be one centimeter. 



MEASUREMENT OF FORCE AND MASS. 1 5 

As g has a value of 980.9 cm. at Paris and the absolute unit of force is 

I 
equal to — of the weight of the unit of mass ai any place whatever, the dyne 

a 

is equal to of the weight (force of attraction to the earth) of a gram 

980.9 

at Paris. And as the value of g at London is 32.191 ft., the poundal is 

equal to of the weight of a pound at London. 

32.191 

36. Gravitation System. — In this system the weight of the pound or 
gram (that is, the force by which the pound-weight or gram-weight is drawn 
toward the earth) is chosen as the unit of force. The weight of the kilogram 
is also frequently used as a unit. 

The mass of a body is therefore measured by its weight in pounds or 
grams or kilograms, divided by the acceleration produced by gravity. 
This quotient is evidently a constant for any particular body; for, if its 
weight W (that is, the force by which it is attracted to the earth) varies 
from any cause (as from change of latitude, change of altitude, etc.), the 
acceleration g will vary proportionally. (Law II.) 

Since in all systems the unit of mass weighs g units of force, it follows 
that the unit of mass in the Gravitation System must weigh g pounds, 
grams, or kilograms, according to which of these is employed as a unit of 
force. That is, its mass must be g times that of the pound-weight, gram- 
weight, or kilogram-weight. 

In the Gravitation System g is usually expressed in feet or in meters. 

The unit of mass in the Gravitation System has never acquired any 
distinct name, although the term Slug has been suggested. When this 
system is used, care must be taken to express the mass of the body in terms 
of this unit and not in pounds or grams. 

We shall see hereafter that the weight of the pound or gram, and con- 
sequently g, vary with the locality, so that the units of both force and mass 
in the Gravitation System are variable unless the definite value of g at some 
particular place is assumed as a standard, which has not been customary. 
This want of definiteness constitutes a fatal objection to the use of the 
Gravitation System where the greatest accuracy is required. It is com- 
monly used, however, in ordinary engineering computations in which the 
variations of g may be neglected. 

The value for ^ of 32.2 feet or 9.8 meters may always be employed in the 
Graviiation System, no closer approximation being necessary. 

The expression for a force in gravitation units may be transformed 
into absolute units by multiplying the numerical value of the force in 
gravitation units by the value of g for the locality at which the measure= 



1 6 MECHANICS. 

merits are made. Methods of determining the value of g for any place 
will be explained later. 

The student will notice that in the discussion of this subject ambiguity 
is likely to arise from the fact that the terms " pound " and " gram " are used 
in two different senses, both as denoting a mass, and as denoting the force 
by which this mass is drawn toward the earth. To remove this ambiguity 
as far as possible, the term "weight of a pound or gram" has been used in 
the preceding discussion whenever a force is referred to. The pound and 
gram are primarily standards of mass, and the use of the same names to 
denote forces is a secondary appHcation. Whenever the terms "pound" 
or "gram" are used as denoting forces, they are to be understood as really 
meaning the weight of the units of mass denoted by those names. 

In the equation W = Mg, which is true for all systems of measurement 
of mass and force, W expresses in units of force the downward tendency 
(due to its weight) of a body of mass M. W and M must always be ex- 
pressed in units of the same system, either Gravitation or Absolute. Also 
in both systems g expresses the number of units of force by which a unit 
mass is drawn to the earth. 

37. Example. — The application of the absolute system may be made 
clearer by a simple example. 

Let us suppose that a certain force A as measured in Paris is equal to 
the weight of 100 grams at that place. A certain other force B as meas- 
ured at London is equal to the weight of 1,000 grams at that place. Since 
the force by which a gram is attracted to the earth is not the same at these 
different locaHties, we cannot compare the forces directly. If, however, 
we reduce the forces to absolute units, the difficulty is avoided. The force 
A expressed in absolute C. G. S. units is A = 100 X 980.9 = 98,090 
dynes, and also B = 1,000 X 981.2 = 981,200 dynes; since the value of 
g for Paris is 980.9 cm., and that for London is 981.2 cm. These results 
immediately show the absolute relation of the forces A and B. 

It is evident that an absolute system may be based upon any con- 
venient units of length, mass, and time. Also a system of measurement 
might be used in which the unit of force was taken as the invariable weight 
of the standard pound or gram at some particular place. 

A table of the units most commonly used will be found on p. 19. 

38. Note* — A philosophical study of the nature of force and of our 
knowledge of it, shows that ultimately we have no way of iiu^a^urinL; llie 
relative mechanical magnitude of all kinds of forces excc})l 1»> their effect 
in generating momentum. Hence, Law 11. of Molion is rather a definition 
of a system of measurement than a law dctermiiuHl from measurements 
obtained by other means. Wc call forces nuThanically i-(|ual when they 
l)rodu(c an equal rate of change of momentum; and if one fonv j)roduces 
a rate of chani^c ;/ times as great as that produced by a second force, we call 
the former force ;/ times as irreat as the latter. 



ENERGY. 17 



TABLE OF DIMENSIONS. 

Quantities Dimensions 

Length L 

Mass M 

Time T 

Area L" 

Volume 1,3 
A 1 Arc 

Velocity = — Zr-^ 

Angular Velocity = ^=- T-^ 

Acceleration = -=- LT-^ 

Momentum = MV MLT-^ 

Force = Ma MLT-^ 

Torque = Fl ML^T-^ 

Energy = JtV/F^ l/L^r-^ 

Moment of Inertia = Mk^ ML" 



Power = J —^^^ ML" T-^ 



ENERGY. 

39. Work and Energy. — Work is performed whenever a force pro- 
duces motion in opposition to a resistance; or more generally whenever 
a force acts to move its point of application through space. 

Work is found to be directly proportional to each of these variables, 
force and distance. Hence representing the force by F and the distance 
through which its point of application is moved by S, the work done is 
expressed by the equation E = FS. 

"Energy is the capacity of doing work." — Maxwell. 

40. Units. — The unit of work in practical use by engineers is the 
kilogrammeter in Metric, and the joot-pound in British measures. The 
kilogrammeter is the work done in overcoming a force equal to the weight 
of one kilogram through the space of one meter. The foot-pound is the 
work done in overcoming a force equal to the weight of one pound through 
the space of one foot. The kilogrammeter is equal to 7.233 foot-pounds. 
The gram-centimeter is frequently employed as a smaller unit of work. 
The foot-ton is often used when great amounts of energ}^ are to be con- 
sidered. 

In purely scientific investigations, absolute units are commonly employed. 
The absolute unit of work is the work done in overcoming an absolute 
unit of force through a unit of length. 



l8 MECHANICS. 

The unit of work in the centimeter-gram-second system is called the 
erg, and is the work done in overcoming a force of one dyne through one 
centimeter. It is a dyne-centimeter. In the foot-pound-second system, 
the unit is the joot-poiindal, which is the work done in overcoming a force 
of one poundal through one foot. 

The rate of work or work-rate is the work done in a unit of time. 

FS 
That is, R = — • If the rate is not uniform, this definition apphes to 

dE 
its average value. In general, R = — — 

Lord Kelvin has given the title "activity" to the rate of doing work. 
It is usually called "power" by engineers. 

A standard in ordinary practical use, for comparing the rate of work 
or the activity of different motors, is the horse-power, which is equal to 
33,000 foot-pounds per minute, or 550 foot pounds (76.0 kilogrammcters) 
per second. One horse-power is equal to 7.46 X lo^, or 7,460 miUion ergs 
per second, assuming g = 981 cm. The French force de cheval or cheval- 
vapeiir, is not identical with the English horse-power, but is defined as 75 
kilogrammeters per second. This equals 7.36 X lo^ ergs per second, if 
g = 981 cm. 

A unit of work frequently used in connection with electrical measure- 
ments is the Joule = 10,000,000 (10^) ergs. The corresponding unit of 
activity is the Watt = lo^ ergs per second. The kilowatt, a unit commonly 
used in rating electric machinery, is 1,000 watts = 1.34 horse-power. 

Engineers often use the horse-power-hottr, the watt-hour, the kilowatt- 
hour respectively as practical units of work. A horse-power-hour is the 
work done in one hour when energy is expended at a constant rate of one 
horse-power. The other units referred to are defined in like manner. 

41. Accumulated Work. Kinetic Energy. — Frequently the space 
S in the formula for work done is not directly known, while the velocity 
with which a body is moving is given. An expression for the work which 
can be done by a body of mass M, moving with a velocity V, may be found 
by ascertaining the space through which the body would move against a 
constant resistance and with a uniformly retarded motion before its velocity 
would be reduced to zero, and substituting this space as expressed in terms 
of V in the general equation. 

If the body is moving with a velocity V against a resistance F which 
is capable of producing a retardation a, the distance over which the body 

will move is 6' = • Substituting this value, wc have E = FS = F 

2a 2a 

F W W V^ 

But — = - , whence/: = = p/F^ 



ENERGY. 



19 



Hence the work which a moving body is capable of performing in virtue 
of its motion is equal to half its mass into the square of its velocity. 

The result is expressed in units of either the absolute or the gravitation 
system, according as the mass is expressed in the one or the other of these. 

It will be observed that in the preceding demonstration it is assumed 
that the work necessary to bring a body to rest is the same, whatever 
may be the nature of the resistance overcome, which is an experimentally- 
determined fact. 

The energy possessed by a body because of its motion is called Kinetic 
Energy. 

The product MV^, which is twice the kinetic energy of a moving mass, 
was called by Leibnitz the vis viva of the mass. 

TABLE OF UNITS. 



System 



Absolute 
Metric 
C. G. S. 



Gravitation 
Metric 



Gravitation 
British 



Mass 


Length 


Time 


Force 


Gram 


Centi- 
meter 


Second 


Dyne 


"Metric 


Meter 


Second 


Kilo- 


Slug" 






gram. 


weighing 








9.8 kg. 








"Slug" 


Foot 


Second 


Pound 


weighing 








32.2 lb. 









Energy or Work 



Erg = Dyne- 
centimeter 

Joule = io7 
Ergs. 



Watt-hour 
Kilowatt-hour 



Kilogram- 
meter. 



Foot-pound 
Horse-power- 
hour 



Power or Activity 



Erg 




Second 




XSJ^iff — 


Joule 




Second 


= Volt 


-ampere 


= 10^- 


Erg 



Second 

Kilowatt = 
1,000 Watts 

Force de Cheval 
Kilogrammeter 



75 



Second 



Horse-power = 
Foot-pound 



550 



Second 



42. Rotating Bodies. — It remains to be shown how the kinetic energy 
of a rigid rotating body is determined. The angular velocity of such a 
body is measured by the angle in circular measure through which it rotates 
in a imit of time, and is numerically equal to the linear velocity of any 




20 MECHANICS. 

point at a distance unity from the axis of rotation. That is, the angular 

d , dd ^, 

velocity oj = — or in general co = — , where is the angle described in / 
/ dt 

seconds. 

Let Wi be the mass of a particle of the 
body situated at a distance r^ from the 
axis of rotation through C (Fig. 5), and 
let CO be the angular velocity of the 
body. The energy of the particle Wj is 
ifUjVj^, or im^r^^oj^, since Vj^ = fjo;. 

The energy of any other particle of 
mass W2 at a distance ^2 from C is Jwarg^^;^. 
Pjc 5^ Similar expressions may be obtained for 

all the particles of the body. Hence the 
total energy of rotation of the body is 

E = ^m^rj^co^ + ^mzrz^oj^ + ^m^rs^oj^ . . . + ^m^/^^o)^ = ^oj^Imr^; 

whence the formula E = ^aflmr^ represents the accumulated energy of 

the body due to its rotation about C, and is the amount of work which 

it will perform while being brought to rest. 

43. Moment of Inertia. — The expression Imr^ is of very frequent 
occurrence in dynamics, and is known as the moment oj inertia of the 
mass. Denoting it by /, we have E = ^io^I. That is, the energy of 
rotation of a body is equal to its moment oj inertia midtiplied by halj the 
square oj its angidar velocity. The quantity / evidently varies with the 
form of the body, its mass, and the position of its axis of rotation. 

Using the notation of the Calculus we may write I = j r^dm. 

44. Radius of Gyration. — If we assume the total mass of a body to 
be concentrated in a single point situated at a distance k from the axis 
of rotation such that the moment of inertia of the mass thus concentrated 
shall equal the moment of inertia of the distributed mass of the body, 
then 



sjL. 

^ M 

The radius k is called the radius oj gyration of the mass. 



Mk-" = / and j^ = , ,^ 

M 



45. Moment of Inertia about any Axis. — Knowing the moment 
of inertia Ig with reference to an axis passing through the center ol gravity 
G of a mass it is often necessary to find the moment of inertia la with refer- 
ence to an axis parallel to the former and passing through a point A at a 
given distance d from G. 

In Fig. 6 let m be any element of mass, ig its moment of inertia relatively 



MOMENTS OF INERTIA. 21 

to an axis G, ia its moment of inertia relatively to an axis through A^ and 

denote by Vg and ra the respective ' 

distances of m from G and A. 

Let GC = X, mC = y. Then 

ig = mf/ = m {x'^ -\- y'^)' Also 

ia = Wfa"" = w \{x + i)2 4- 3;2j 

whence ia — ig = w {2dx + rf^) 

= 2 rf m X + w ^^. For any 

other element m\ i'a — i'g = 

(2 d m' x' + m'd^), and for the Pig ^ 

whole mass Jia — ^ig = 2 dl mx 

+ d^Im. But I m X = o, since this is the sum of the values of mx for 

all the particles composing the body with reference to an axis through the 

center of gravity. Hence la — Ig = Md^ and h = Ig + Md"". 

46. Determination of Angular Velocity.— The angular velocity 
generated by the expenditure of an amount of work FS upon a body can 
be found as follows: — 




FS = ioj^I; whence co = 



v/^. 



It will also be seen that by solving this equation relatively to I an expression 
may be found for the moment of inertia, thus indicating a way in which 
the value of this can be determined experimentally. 



TABLE OF MOMENTS OF INERTIA. 

From Routh's Rigid Dynamics. 

The moment of inertia of 
(1) A rectangle whose sides are 2a and 2b, 
about an axis through its center in its plane 

perpendicular to the side 2a — mass X — » 



3 

a^ + Z>2 



about an axis through its center 

perpendicular to its plane = mass X 

6 
(2) An elHpse, semi-axes a and b, 

about the major axis a = mass X — > 

4 

about the minor axis b = mass X — > 

4 

about an axis perpendicular to its plane 

through the center = mass X 

4 

In the particular case of a circle of radius a, the moment of inertia about 

a diameter = mass X — , and that about a perpendicular to its plane 

4 

through the center = mass X 



22 MECHANICS. 

(3) An ellipsoid, scmi-axcs (/, /), r, 

b' + r- 

about the axis a = mass X 



5 

In the })articular case of a sphere of radius a the moment of inertia about 

1- 2 

a diameter = mass X — ci^- 

5. 

(4) A right soHd whose sides are 2^, 2/', 2r, 

about an axis through its center perpendicular ^ _ ^ w ^^^ + ^^ 
to the plane containing the sides h and c \ ~ ' " ^ 

These results may all be included in one rule, as an assistance to the 
memory. 

, , . . . . (sum of squares of perpendicular 

Moment of mcrtia semi-axes) 

about an axis >• = mass X • 

of symmetry ) 3^ 4 or 5 

The denominator is to be 3, 4, or 5 according as the body is rectangular, 
elliptical or eUipsoidal. 

47. Kinetic Energy. — Energ}- is either Kinetic or Potential. Kinetic 
energy, as already explained (see § 41, p. 18), is the capacity possessed by a 
body of doing work in virtue of its motion, and is expressed by the equation 

K = iMV\ 

Illustrations: Projectile in motion; fly-wheel; stream of flowing 
water actuating undershot water wheel; current of air moving wind-mill. 

The term "kinetic energy" is due to Kelvin. It is sometimes called 
"actual energ}', " as proposed by Rankine. 

48. Effect of Unbalanced Force on Mass. — If an unbalanced force 
acts to accelerate the motion of a mass, its effect is to add to the kinetic 
energy of the mass. The increment of kinetic energy, that is, the work 
stored up, in a given time, is 

K = WiV^ - TV). 
If the mass, being in motion, moves against a resistance, it does work 
which is measured by the decrease of kinetic energy; that is, 

K = p/(IV - T\^). 

The increase oj kinetic energy produced by the cojitinucd adion oj a force 
on a mass is equivalent to the jorce multiplied by the space through luhich it 
nets. This will aj)pear from the following demonstration: — 

Consider the action of the constant force F on a ma.ss M through a sjKice 
S. l^hen as F = Ma and S =^ Vjt ^- \ at\ we have FS = ^fVo at -f \ MaH\ 
This is e(|ual to the gain of kindif iMUTgy in passing through .S". l'\)r 
(•.illing A'oand A', the kinetic (Micrg\- a1 tlu' beginning and end of .V resjiect- 
ively A'o = JA/IV, A', = \M{Vo + <^0' whence the gain in kinetic energy, 
A', — A'o = MVq at -\- \Ma^t^ which has already been shown to be the 
value of FS. Hence FS = h ^t{V.' - ^V)• 



EFFECT OF UNBALANCED FORCE. 



23 



If the force F is not constant during the time t, we may divide t into a 
very great number of parts so small that during each one of them the force 
acting may be considered as constant. Then, for each one of the elementary 
spaces traversed in an element of time, the proposition will hold. Hence 
we have for their sum in all cases 

Fds = i M{Vi^ - Fo^) 

So 

This equation may be derived directly by the methods of the Calculus 
as follows: — 

FdS = Fvdt = Fatdt = (Ma) at dt. Hence I FdS = j MaHdt = 

_ _ t/ 5n tJ to 



111 a 

J; 



MaH^ 



. Hence F(S, - S^) = JM(F,^ - F^^). 



The demonstration evidently applies whether the force F is constant or 
variable. 

From a similar mode of reasoning, it follows that when a moving mass 
M overcomes a resistance F through a space S, the work done by the 
body results in a loss of kinetic energy as expressed by the equation 

FS = iM(^o^ - Vx')- 
In the first case considered FS is commonly spoken of as the work done 
upon the body by the force F in time T. In the second case the work is 
done by the body against the force F. 

49. Total Kinetic Energy. — The total kinetic mechanical energy 
of a body is the sum of the energy due to translation and that due to ro- 
tation; i. e.y 

Kt = iMF- + WJ- 
For a system of bodies the sum of the energies of each part must be 
taken. 

50. Equivalent Mass. — It will be seen that if a force acts on amass 
to produce simultaneous translation and rotation, as, for example, when a 
ball is caused to roll on a surface, the relation between the work done upon 
the body and the total resulting kinetic energy is represented by the equation 
FS = iMV^ + ico^I. 

If we denote by Me a mass such that with the linear velocity F alone it 
would possess an amount of energy equal to the total energy of M, we 

have 1 MeV = i MV' + W^- As F = coR, Me = M + -^' 

Thus a sphere descending an inclined plane moves with a linear accel- 
eration only f of that which it would acquire did it not rotate, since for it 
Me = iM. 



24 MECHAXICS. 

51. Potential Energy. — This is the capacity of doing work possessed 
by the bodies or particles composing a system in virtue of their relative 
position. If bodies are so situated that they are acted upon by a force 
which will produce motion in them on the removal of some restraining force, 
and thence generate kinetic energy, the system is said to have potential 
energy. Thus a mass suspended at an elevation will fall as soon as the 
cord sustaining it is cut. The potential energy in this case is P = FS, 
where F is the weight of the raised mass, and S is its elevation; or P = 
J MV^ where V is the velocity corresponding to a fall from the height S. 

Additional Examples: Stretched spring; head of water. 

In general the potential energy of a system is measured by the work 
expended in reducing its energy to zero; or conversely, by the work which 
must be done to endow the system with the amount of potential energy 
which it possesses. 

Potential energy is sometimes called ''energy of configuration or strain." 
The term "potential energy" was originally suggested by Rankine. 

52. Transformation of Potential Energy into Kinetic, and the 
Reverse. 

Examples: Pendulum; tuning-fork in vibration; hydraulic ram. 

53. Conservation of Energy. — The sum of the kinetic and po- 
tential energies oj a system of bodies not acted upon by any external force is 
a constant. Or, in other words, the total energy oj a system oj bodies is not 
affected by their mutual actions. 

This fact is learned from observation and experiment, and is found to 
be true without exception. 

As an illustration consider the case of a body falling from a height h^ 
and which has descended to a height hj. The potential energy is \Vh^ 
and the kinetic energy is \MVi^ = WQi^ — h^). The sum of these is 
evidently a constant and equal to the original potential energy Who, or to 
the final kinetic energy on reaching the earth. 

The same reasoning can be applied to the case of a body thrown upward, 
a pendulum, a vibrating tuning-fork or string, etc. 

54. Conservative System. — "When the nature of a material system 
is such that if, after the system has undergone any series of changes, it is 
brought bark in any manner to its original stat(\ the whole work done by 
external agents on the s\sleni is e(jual to the whoK' work i\n\]c \)\ the svstem 
in overcoming external force, the system is called a < nnscr:\iti:r .vy.v/r;;/." 
— Maxwi.i.i.. 

55. Different Forms of Energy. — lliere are various forms of energy 
other than mechanical energ\-, th{> only kind which wi^ hav(> thus far con- 
sidered. Several schemes of classilieation of these* ]ia\-e been suggested. 
Tile following is a modification of one given long since b\" IVofi'ssor 
Ha hour Stewart: — 



FORMS OF ENERGY. 25 

I. Visible Mechanical Energy. 

a. Visible kinetic energy. 

b. Potential energy of visible arrangement. 
II. Invisible Molecular Energy. 

a. Kinetic energy of absorbed heat. 

b. Radiant energy (kinetic). 

c. Potential energy due to molecular separation by heat. 

d. Potential energy due to chemical separation. 
III. Energy of Electricity. 

a. Potential energy due to electrical charge. 
. b. Kinetic energy due to electrical current. 

Another mode of classification possessing certain advantages is the 
following, given by Professor A. A. Noyes: — 

1. Kinetic Energy. 5. Electrical Energy. 

2. Gravitation Energy. 6. Magnetic Energy. 

3. Cohesion Energy. 7. Chemical Energy. 

4. Disgregation Energy. 8. Heat Energy. 

56. Note. — It is possible that all energy is in fact kinetic, that potential 
energy, so-called, is always due to undetected motion of some kind, though 
of an unknown nature. While this has not as yet been proved to be the 
case, it is a plausible hypothesis. 

For example, Kelvin has shown that the phenomena of elasticity and 
hence potential energy of strain may be accounted for by an assumed ro- 
tation of elementary portions of a mass. Again, the explanation of gravi- 
tation by Le Sage assumes the bombardment of masses by " ultra-mundane 
corpuscles." 

A familiar example of a case in which potential energy possessed by a 
mass is in fact due to motion is found in the case of a gas, which is capable 
of doing work in virtue of its expansive tendency or pressure. But, accord- 
ing to the Kinetic Theory, this pressure is caused by the rapid motion of 
the molecules of the gas. The potential energy of the gas is really the kinetic 
energy of its molecules. 

57. Energy Transformations. — Any one of the various forms of energy 
may give rise, either directly or indirectly, to any other form. 

Illustrations: Heat produced by percussion or friction; steam- 
engine: electrification by friction; attraction and repulsion of charged 
bodies: dynamo-electric generator; electric motor: thermo-electricity; 
wire heated by electricity: heat of chemical combination; dissociation by 
heat: voltaic battery; electro-chemical decomposition. 

The above proposition relates only to the qualitative aspect of the differ- 
ent transformations of energy. It was formerly known as the principle of 



26 MECHANICS. 

the "Correlation of Forces," a term introduced by Grove in 1843, ^^ which 
early date the distinction between force and energy was not clearly recog- 
nized. 

58. Conservation of Energy.— The following law, which is a state- 
ment of the doctrine of the conservation of energy, holds for all known forms 
of physical energy. '' The total energy of any body or system of bodies is 
a quantity which can neither be increased nor diminished by any mutual 
action oj these bodies, though it may be transformed into any of the forms 
of which energy is susceptible." — Maxwell. 

59. Illustrations of Principle.— For example, suppose a dynamo- 
machine to generate a current of electricity which in its turn drives an 
electric motor. If the work done in driving the dynamo is measured, and 
also the heat generated mechanically by friction of machinery, resistance of 
air to moving parts, etc., the total heat generated by the current in the 
complete circuit, and the total mechanical w^ork done by the motor, then 
the mechanical work done in driving the generator will equal the total 
amount of energy developed in the several resulting operations. 

Again, if a definite weight of zinc in a battery cell is oxidized and the 
chemical energy disappearing in the cell is wholly transformed into the 
electrical energy of the current produced, and this current expends its energy 
in heating a resistance coil, it will be found that the same amount of heat 
is liberated in the circuit that would have resulted from the direct oxidation 
of the same weight of zinc in a calorimeter. 

60. Case of Heat. — When a change of state occurs on raising the 
temperature of a body, there is always an absorption or liberation of 
heat, corresponding to the expenditure of work (positive or negative) 
accompanying the change. Thus, when a liquid is caused to assume 
a vaporous condition, a great amount of heat disappears {latent heat of 
vaporization). This is due to the fact that mechanical work is done by 
the expenditure of heat, first in separating the particles of the liquid 
from one another in the act of evaporation or of boiling {internal 
work), and second, in overcoming the atmospheric pressure which re- 
sists the expansion which accompanies the change of state {external work). 
When ice is melted, the large amount of heat that disappears {latent heat 
of fusion) is chiefly expended in doing internal work. Increase of either 
the internal or external w^ork tends to increase the latent heat. Increased 
external pressure raises the fusing-point of substances which expand on 
melting because of the resulting increase in the external work which must 
be performed. An opposite effect occurs with substances which like ice 
contract on melting. Anything which increases the internal work accom- 
panying a change of stale, as, for example, in va]:)orization, raises the boiling- 
point. Thus we may suppose that the rise of the boiling-point of liquids 



ENERGY FACTORS. 27 

when solids are dissolved in them is due to the increased difficulty of sepa- 
rating the liquid from the dissolved solid. 

In ordinary evaporation the heat which necessarily disappears on ac- 
count of the change of state is withdrawn from neighboring bodies, thus 
lowering the temperature to a greater extent according as the evaporation 
is more rapid. 

61. Mechanical Equivalent of Heat. — The quantity of mechanical 
work which is capable of generating one unit of heat, is called the mechanical 
equivalent of heat, or Joiile^s equivalent, from the physicist who first deter- 
mined its value. 

The thermal unit commonly employed by engineers in Great Britain 
and the United States is the amount of heat necessary to raise one pound 
of water through i^ Fahrenheit. Owing to the variation of the specific 
heat of water with temperature the temperature of the water must be stated. 
The C. G. S. unit of heat is the gram-degree, the temperature being meas- 
ured in Centigrade degrees. The readings are ordinarily reduced to the 
standard hydrogen thermometer. 

Several determinations later than that of Joule have been made by 
different persons using various methods. The value obtained by Row- 
land at Baltimore in 1879 with certain corrections subsequently appHed, 
for the temperature 15° C. is J == 778 foot-pounds at Greenwich = 4.19 X 
io7 (41,900,000) ergs. 

62. Energy Factors. — It is found that with all forms of energy two 
factors, upon both of which the quantity of energy depends, must be taken 
into consideration. One of these is called the capacity factor, the other the 
intensity factor. The energy is found to depend upon the product of these. 

For example, in the expression for mechanical kinetic energy, K = ^MV^, 
M is the capacity factor, V^ the intensity factor. In the case of the energy 
of a compressed gas the volume is the capacity factor, the pressure the 
intensity factor. In the case of heat energy, the heat-capacity at a certain 
temperature is the capacity factor, the temperature the intensity factor. 
In the case of an electric charge, the electrical quantity is the capacity 
factor, its potential the intensity factor. 

It is the intensity factor which determines the tendency towards change 
of energy either as to distribution or form. It is because of the pressure of 
compressed gas that it tends to expand. The temperature of a mass de- 
termines the tendency towards a transfer of heat; the potential of an electric 
charge determines the tendency toward a flow of electricity. 

The intensity factor is a relative quantity. Thus gaseous pressure, 
temperature, electric potential are magnitudes relative to some value taken 
as a standard. Hence neither transfer nor transformation of energy can 
take place unless there exists a difference in the intensity factor between 



28 MECHANICS. 

different parts of the system. Thus there is no transfer of gas between two 
connecting reservoirs if the pressure of the gas is the same in each; there is 
no resultant transfer of heat between two bodies whose temperature is the 
same; nor of electricity between two bodies at the same potential. 

63. Function of Machines. — It will be seen from what precedes that 
the function of a machine is merely to transfer or transform energy, not to 
create it. 

Neglecting friction and other harmful resistances, the effective force 
applied to any machine multiplied into the distance through which its 
point of application is moved in the direction in which the force acts, is 
equal to the resistance overcome multiplied into the distance through which 
it is overcome. That is, the mechanical work supplied to the machine is 
equal to the mechanical work done by the machine in overcoming resistance. 

On account of friction and various other harmful resistances, the latter 
of these is in practice always less than the former. 

Examples: Lever, wheel and axle, pulley, and other "mechanical 
powers." 

The efficiency of a machine is the ratio of the work which it does to the 
work done upon it. 

64. Measurement of Work of Machinery. — The mechanical energy 
expended in driving a machine may be measured by different forms of 
dynamometer or erg-meter. 

I. Friction Brake. The 

shaft (Fig. 7) is made to run 

TTk] at its normal speed, overcom- 

"'' ~ ing the friction of the brake as 

a resistance. Calling this 

Pj^j ^ speed n (r.p.m.), w the 

balancing weight in the 

brake-pan, a the arm of the brake, the work done by the machine is 

27rnwa 

W = w a X 2nn per minute, whence H. P. = • , if w and a are ex- 

33000 

pressed in pounds and feet respectively. 

2. Transmission Dynamometer. The work done is measured under 
actual conditions of use. In the torsion dynamometer the torque is deter- 
mined from the twist of a shaft or helical spring through which the power 
is transmitted. 

65. Perpetual Motion. — ^In its technical meaning, a "perpetual- 
motion machine" is not a machine which would run indefinitely if its 
working parts remained unimpaired, but a machine which does work with- 
out the expenditure of energy. 




I 
I 

L 




SOURCES OF ENERGY. 29 

Thus a windmill operated by the trade-winds, or a turbine operated 
by Niagara Falls, or an engine working automatically by the rise and fall 
of mercury in a thermometer or barometer tube would not be a perpetual 
motion machine, since energy would be expended in its operation. Nor 
is the revolution of a planet about the sun perpetual motion since no work 
is done in the revolution. 

It is a logical consequence of the Principle of the Conservation of Energy 
that "perpetual motion" as above defined is an impossibility. 

66. Sources of Energy. — Tait. 

a. Potential. 

1. Fuel. 

2. Food of Animals. 

3. Ordinary Water Power. (Head.) 

4. Tidal Water Power. 

b. Kinetic. 

5. Winds and Ocean Currents. 

6. Hot Springs and Volcanoes. 

Immediate Sources. 

1. Primordial Energy of Chemical Affinity. 

2. Solar Radiation. 

3. Energy of Earth's Rotation. 

4. Internal Heat of Earth. 

67. The Sun as a Source of Energy.— Substantially all the energy 
which is utihzed by man is derived from the sun, e. g., that of water power, 
wind power, fuel, food, electrical power, though not that of tidal water 
power. 

The heat entering the earth's atmosphere from the sun according to the 
determination of Abbot (1908) is 2.1 calories (gram-degrees) per square 
centimeter per minute. This is known as the so/ar co7istant. Hence 
energy is received at the rate of 0.15 X lo^ ergs per second = 0.15 watts. 
This corresponds to radiation from each square centimeter of the sun's 
surface at a rate of about 9 H. P. The temperature of the sun's surface is 
probably in the neighborhood of 6,000° to 7,000° C. 

Helmholtz (1854) explained the origin of this energy as due to slow con- 
traction of the gaseous sun. A yearly diametral contraction of not over 
250 feet would suffice to produce the observed amount. Assuming the 
sun to have contracted from a sphere filling the orbit of Neptune the calcu- 
lated possible radiation would not suffice for over 18,000,000 years of emis- 
sion or thereabouts. On these data Kelvin has based an estimate of the 
maximum possible "age of the earth." Recent discoveries within the 
domain of radioactivity have somewhat disturbed these conclusions. 

68. Historical.— The truth of the doctrine of the Conservation of 
Energy in its complete form came to be recognized very gradually, prin- 
cipally during the first half of the nineteenth century. No one person can 



30 MECHANICS. 

be named as its originator. Newton was apparently cognizant of the 
principle so far as concerns machines, and recognized the fact that there 
could be no gain of power by such means. The great French masters of 
applied mathematics in the latter part of the eighteenth century, D'Alem- 
bert, Lagrange, Laplace, and others, assumed its truth so far as concerns 
mechanical forces, and based their most important generalizations upon it. 

Early in the nineteenth century Dr. Thomas Young introduced the 
term "energy" to denote mechanical energy or more strictly vis-viva. 

In 1798 Benjamin Thompson, Count Rumford, from a study of the 
enormous amount of heat developed in boring a cannon showed conclusively 
that heat could not be an "imponderable fluid," as was then beheved, but, 
on the contrary, is a "mode of motion," and produced by friction. It 
was not until a much later date, however, that this view came to be the 
accepted one. 

Rumford 's views were confirmed and adopted by Sir Humphry Davy 
(1799) and Dr. Thomas Young (1807), but beyond these they met with 
slight acceptance. Even Fourier (1822) and Carnot (1824) accepted the 
"caloric" theory of the material nature of heat. 

Seguin in France (1839), Mayer in Germany (1842), Colding in Den- 
mark (1843), and Helmholtz in Germany at about the same time, all 
adopted the view that heat is a form of what we now call energ)', and en- 
deavored to ascertain the laws of its generation and transformations. 
Mayer showed a way of computing the mechanical equivalent of heat from 
already known data, which though subject to criticism from a logical point 
of view nevertheless proved capable of furnishing approximate results. 

Joule in England, whose work began as early as 1840, made a series of 
contributions to this branch of science which are of inestimable value. 
In 1843 he showed that the heat produced in a given circuit by a given 
quantity of electricity generated by a magneto-electric machine was the 
same in amount as that which would be produced by direct transforma- 
tion into heat of the work consumed in producing the current of electric- 
ity by the machine. He also obtained from his measurements a value for 
the mechanical equivalent. Shortly thereafter he determined this constant 
by direct measurement under different conditions of the work done by fric- 
tion and of the resulting heat. Using various methods of producing heat 
by the expenditure of mechanical work, all of which were found to give a 
like value of the mechanical equivalent, he showed this to be independent 
of the particular manner in which the work was done. In 1850 he pub- 
lished a paper of great completeness giving a value of J = 772 foot-pounds 
which continued to be accepted for almost thirty years. 

In 1847 Joule in Manchester and Helmholtz in Berlin each read a paper 
now regarded as epoch-making in iniportance, in which the principle of 
the conservation of energy in its full breadth was clearly set forth. But 
their views were not accepted in cither country, and Poggendorflf's Aniialoi, 
the leading physical journal in Germany, declined to publish the pa|X}r of 
Helmholtz. But largely through the ability and influence of William 
Thomson (afterwards Lord Kelvin) and Du Bois Reymond, the new views 
rapidly gained acceptance. 

The reverse transformation of heat into mechanical work was investi- 
gated by Hirn, Rankine, Kelvin, Clausius, and others, showing the same 
(quantitative relationship to exist between heat transformed and the resulting 
work. 



DISSIPATION OF ENERGY. 3 1 

The expression "Conservation of Energ}^," replacing the earher and 
confusing expression "Conservation of Force," was introduced by Ran- 
kine. 

The laws of relationship of electrical quantity and chemical decomposition 
were discovered by Faraday (1833), those relating to the heating effects of 
electricity by Joule (1841). More recently the laws relating to heat pro- 
duced by chemical action have been determined by Favre and Berthelot. 

All the results obtained, without exception, have been such as to confirm 
and extend the conclusion that energy is neither created nor destroyed in 
any of its varied transformations. 

69. Dissipation of Energy. — All energy tends to pass from a higher 
form to a lower one, and ultimately to assume the form of uniformly diffused 
heat. — Kelvin. 

Hence, although the total energ}^ of a system remains constant, the 
amount of its available or free energy continually diminishes toward zero. 
Thus it is easy to transform completely the total amount of mechanical 
energy of a system into heat energy, but it is impossible to transform the 
total amount of heat energy of a system into mechanical energy. The 
total electrical energ}^ of a system can be transformed into heat, but the 
total heat energy of a system cannot be transformed into electrical energy. 

For example, it can be shown that with a perfect steam or other heat 
engine, if the vapor enters the cyhnder at a temperature of 200° C, and 
leaves it at a temperature of 110° C, the efficiency can be only 19 per cent. 
With the actual engine it is much lower. 

70. Application of Principle to Cosmical Phenomena. — The 
principle of the dissipation of energ}' applies to the physical universe as 
far as we know it, whence wt conclude that the available energy of the 
universe tends toward zero. There is, however, no evidence of retardation 
of planets or comets such as w^ould be caused by a resisting medium in space. 

71. Transmission of Energy — i. Transference of body possessing 
energy. 2. Waves. 

72. Waves. — In any wave the wave-form alone progresses, while the 
particles at any point merely assume an oscillatory or rotary motion. 

Waves are of three kinds : longitudinal, transverse, or torsional, according 
as the motion of the particles is parallel to the line of transmission of the 
wave, transverse to it, or twisting about it as an axis. 

The following classes of wave may be produced in ordinary matter: 
(a) Gravitational waves in liquids, due to displacement of level, {h) Ca- 
pillary waves or surface ripples, due to surface tension of the liquid, {c) 
Pressural waves in body of sohd, Hquid or gas, due to volume elasticity. 
{d) Distortional waves in sohd, due to elasticity of shearing, bending, 
twisting. Of these (a) and {h) are necessarily transverse waves, (c) longi- 
tudinal, {d) either transverse or torsional. 



32 MECHANICS. 

Water waves are transverse; the sound wave is longitudinal. 

In the ether, electromagnetic waves exist, of which light waves and the 
electric waves, utilized in space telegraphy, are examples. These are due 
to electrical and magnetic properties of the ether which are analogous to 
elasticity and mass in ordinary matter. Electromagnetic waves are trans- 
verse in their character. 

CURVILINEAR MOTION. ROTATION. 

73. General Principles.- Definitions.— Curvilinear motion is pro- 
duced by the continuous action of a deflecting' force upon a moving particle 
or mass. The deflecting force may vary in any manner whatever, both as 
to magnitude and direction. 

An important case is that in which the deflection is always toward the 
same point, in which case the deflecting force is called a centripetal jorce. 
Such a point is called a center oj jorce. The so-called " tangential or pro- 
jectile force " in reahty is not a force, but the momentum of the particle. 

It will readily be seen that the nature of the orbit traversed by the particle 
will depend upon the manner in which the magnitude and direction of the 
deflecting force vary. Hence from the form of the orbit the law of varia- 
tion of the deflecting force can be determined. 

Newton showed that if the orbit of a body is a conic section the line of 
action of the deflecting force must always pass through a focus of the curve, 
and furthermore must vary in magnitude from point to point of the curve 
in the inverse ratio of the square of the distance of such point from that 
focus; and conversely. This is the case with all members of the solar 
system. All the planets move in ellipses of small eccentricity with the 
sun situated in one focus of the ellipse. Many comets also move in orbits 
which are elliptical, but of great eccentricity. Most comets move in sen- 
sibly parabolic orbits, and some, apparently, in hyperbolic orbits. 

Since the centripetal force acts to draw the revolving body inward toward 
the center of force, it follows from the 3rd Law of Motion that this center 
sustains a radially-outward pull equal in magnitude to the centripetal force. 
It is this tendency of the center of motion to move radially outward which 
led to the conception of the definite existence of a centrijiigal force. In 
fact, there is no such force. It is only the reaction on the axis of the cen- 
tripetal force, which pulls the body and the center equally towards one 
another, though the fixation of the center prevents this from moving. 
The revolving body always tends to move tangentially in virtue of its 
inertia. 

As the tangential momentum, however, tends to increase the radius of 
revolution, the term "centrifugal force" has come to be habitually used 



CIRCULAR ORBIT. 



33 



by engineers and others as denoting a component of the momentum which 
is opposite to the centripetal force, and, necessarily, equal to it. 

74. Centripetal Force in Circular Orbit. — The case most 
frequently occurring in the ordinary applications of Physics is that of a 
particle or body revolving in a circle about a center of force situated at 
the center of the circle. The curvature of the circle being constant, the 
centripetal force in such an orbit is of constant intensity. Denote it by F, 

To determine its value let C (Fig. 8) be the center of the circle in whose 
circumference the particle revolves, PR the 
space over which the particle would pass in 
time t in virtue of its tangential velocity, PM 
the space which the particle would traverse in 
the same time in virtue of the action of the 
centripetal force alone. Let PQ be the chord 
of the arc in which the particle moves. From 
the geometry of the circle, PQ" = PM X PD. 

Denote by / the centripetal acceleration, and 
by V the constant orbital velocity of the particle. 
If s is the space which would be traversed toward 
C in time t under the influence of the centrip- 

d's . ^ _ PQ' 




Fig. 8. 



etal force, 



/. But s = PM = 



dp ' PD 

arcs, the chord and arc coincide, whence PM = s = 

d^s v^ - v' 

/. Hence F = Mf = M- 



Also for indefinitely small 



vH^ 



2R 



and 



dt^ R R 

Hence, the centripetal force varies directly as the mass of the particle and 
the square of the velocity, and inversely as the radius of the orbit. 

Evidently M may be written in either gravitation or absolute units, in 
which case F will be expressed in similar units. 

The centripetal force of any extended body or system of bodies is the 
same as if the total mass were concentrated at its center of mass. Hence 
the preceding demonstration can be appHed immediately to such cases. 

The formula may also be put into the following form: — 

27cR 

Let T be the time of revolution; then vT = 271R, and v = 



Hence F 



M — = M 
R 



^n'R 
2^ 



T 



Hence, for bodies revolving in circular 



orbits, the centripetal force varies directly as the mass and the radius of the 
orbit, and inversely as the square of the time of revolution. 



34 



MECH-A.NICS. 



75- Velocity in Orbit.— The uniform velocity with which a body 
revolves in a circular orbit is equal to that which the centripetal force 
would generate by its constant action upon the body through half the 
radius of the orbit. For 

v^ ■ — ■ 

/ = -~- ; whence v = \/jR. 
K 

Suppose the centripetal force to impel the body from a state of rest until 
it attains the velocity v. Call s the space described. Then 

V = \'^2Js\ whence \/jR = \/2Js, and s = — . 

2 

76. Body Revolving in Vertical Orbit. — That a body may 
revolve in a vertical circular orbit, its centrifugal force must at least be 
equal to its weight. To find the minimum velocity requisite for this, put 
W = F in the general equation. Then 

Wv^ — 

W = — ^, and V = \/gR' 



Illustration: 



Centrifugal Railway. 





77. Practical Illustrations — Cen- 
trifugal dryer, blower, pump, cream- 
separator; centrifuge; bursting of fly- 
wheels; vehicle on curved road (Figs. 
9 and 10); depression of inner rail on 
curve. 



Fig. lo. 



78. Conical Pendulum — In the conical ])cndulum the ball P must 

assume such a position that the cord AP is in the line of action of R, the 

resultant of the weight and the centrifugal force 

of the ball. 

F r 
Under these circumstances tan = „7 = —— , 

\\ h 

denoting l)y ;- the radius of the circle of revolution 
of P and by h the altitude of the cone described 
by the suspending cord. The time of rotation is 




Fig. II. 



J 



7- 



I'or the centrifuiial force 7*' ^ \V - — 

h 



and T 



■V 



Api'LTCATTONS. — Watt's governor for st(\im en- 
gine; regulating du'c liaiii-ni for chronographs; 
(IriN'iiiL^ (lock for ((luatoiial 1c1c>coih\ 



EFFECT OF ROTATION OF EARTH. 



35 



79. Equilibrium Surface of Liquid 
Axis. — This is a paraboloid of revolution, 
liquid surface that surface must be at right 
angles to the resultant of the weight W of 
the particle and its centrifugal force F. 



rotating about Vertical 

At every point on the free 



W 
Hence (Fig. 12) tan 6 = —^=W -v- 



W4n^r 



which value is 




— ; and r tan (9 = - — 

constant for all points on the liquid surface. 
Hence r tan Q = AC^ the subnormal of the 
curve, is a constant, which is a well-known 
property of the parabola. 



80. Centrifugal Force at Equator, — The magnitude of the centri- 
fugal force at the equator due to the earth's rotation is found by substituting 
the proper values of R and T in the general equation. 

The equatorial radius of the earth = 6378250 meters (Clarke, 1880), 
and the value of g at the equator as measured is 9.78 meters. Hence 

v"^ /^Ti'R 4 X (3.i4it;Q3)'' X 6378250 I 

R gT^ 9.78 X (86164)2 288.4 

Hence the effect of the centrifugal force at the equator is to diminish the 
weight of every body by its ^Jg part. Since F varies as v^, a velocity of 
approximately 17 times the present would generate a centrifugal force at 
the equator entirely counterbalancing the action of gravity. 

81. Variation of Centrifugal Force with Latitude. — It will 
appear from a consideration of Fig. 13 that the value of the centrifugal 





Fig. 13. 



Fig. 14. 



force In latitude ^ represented by PF, will be to Its value at the equator 
in the ratio of the respective radii of revolution of a point in these latitudes. 
Hence F) =Fa cos ^. 



36 MECHANICS. 

The diminution of weight due to centrifugal force in latitude X is rep- 
resented by the normal component PN in the diagram. CaUing this A^, 
we have N = F-^ cos ^ = Fq cos^ A. 

It will also be seen that the existence of a tangential component, PT, 
of the centrifugal force acts to produce a deviation of the plumb-line 
toward the equator. For latitude 45° the value of this can be shown to be 
11' 30''. 

Other important phenomena in which the centrifugal force due to the 
earth's rotation is concerned are those of cyclones and storms in general, 
and the tides. A suggested deviation of river courses is doubtful. 

82. Effect of Centrifugal Force on Figure of the Earth. 
Spheroid of Equilibrium — The tangential component PT acts to 
produce motion in a movable body, as, for example, the water of the ocean, 
from the poles toward the equator. Hence with a fluid mass, such as the 
earth must have been at a former remote period, there would result a heap- 
ing up of matter in the equatorial regions, so that the eartn thus rotating 
could no longer be spherical. 

The actual form assumed by such a fluid rotating mass must be such 
that the surface at every point is normal to the resultant of the weight 
of the particle and the centrifugal force. If P, Fig. 14 (p. 35), is any such 
point, PR, the resultant of the weight P\V and the centrifugal force PF 
must be normal to the surface at P. 

Maclaurin (1740) showed that an oblate spheroid fulfils this condition. 
It has since been shown by Jacobi and Poincare that there are several 
other possible figures of equilibrium. 

From the mass of the earth and its period of rotation it is possible to 
calculate the amount of the polar flattening. 

The ellipticity of the earth as measured is -g^. That of Jupiter, whose 
period of revolution is only 9 h., 55 m., is y^. 

83. Plateau's Experiment. — Sphere of oil immersed in mixture of al- 
cohol and water of same density, becomes spheroidal when rotated, and 
finally throws off equatorial rings if the velocity is increased. 

84. Theories of Evolution of Solar System.— (a) Nebular Hypo- 
thesis of Kant (1755) and Laplace (1796). Difficulties in the way of its 
acceptance, {b) Planetesimal or Spiral-Nebula Hypothesis of Chamber- 
lain (1905). 

85. Kelvin's Estimate of <<Age of Earth.''— Ellipticity (assumed 
to be substantially unchanged since soHdification) in connection with in- 
crease in length of day of 22 seconds per century from tidal retardation 
leads to the conclusion that some 20,000,000 years have elapsed since solid 
crust was formed. Estimates based on an entirely different class of data 
lead to a value of the same order of magnitude, as has already been stated 
in connection with the energy of solar radiation. 



ROTATION OF RIGID BODIES. 



37 



Objections. — Plasticity of earth has probably allowed a subsequent 
change in ellipticity. Amount of retardation is not certain, 

86. Rotation of Rigid Bodies — The effect of an unbalanced 
centrifugal force is two-fold, tending (i) to shift the axis of rotation as 
a whole, i. e., to produce a translatory motion 
of the axis, and (2) to produce angular de- 
viation of the body. (See Figs. 15, 16.) 

87. Free Axes. Principal Axes. — 
It will be seen from the figures that, if the 
axis of revolution passes through the center 
of mass, the tendency to translatory 
motion disappears. That there may also 
be no tendency to angular motion, the axis 
of rotation must coincide with an axis of 
symmetry of the body. (See Fig. 17.) If 

the axis of rotation is parallel to an axis of symmetry, but does not pass 
through the center of mass, there will evidently be no tendency to angu- 
lar deviation. An axis about which a body may revolve without causing 
any tendency to angular deviation is called a principal axis. Any axis 




Fig. is. 




Fig. 16. 



Fig. 17. 



about which a body may revolve without producing a tendency to either 
angular deviation or translation of the axis of rotation is called a free axis. 
The free axes are evidently principal axes passing through the center of 
mass. 

In an ellipsoid with three unequal axes, these are the free axes. In 
a right elliptical cylinder the free axes are the axis of the cylinder and 
the major and minor axes of the elliptical middle section of the cylinder. 
Any diameter of a sphere is a free axis; also, any diameter of the equator 
of an oblate or prolate spheroid, together with its polar axis. 

Practical Applications in Machinery: Balancing of fly-wheels 
and other rotating pieces; balancing of crank on driving-wheel of loco- 
motive. 

88. Axis of Stable Rotation.— An inspection of Fig. 16 will also 



38 MECHANICS. 

show that a body is in stable equihbrium only when rotating about its 
shortest diameter. 

In machinery several practical considerations frequently require that a 
rotating piece shall be rotated about a longer rather than a shorter axis of 
symmetry. This, however, is not a case of free rotation, and the rigidity 
of the shafting is made such as to counterbalance any centrifugal couple 
that may be generated. 

89. Analogies between Translatory and Rotary Motion. — Some 
further characteristics of the rotary motion and energy of rigid rotating 
bodies are most readily considered in this connection. 

It is easily seen that the effect of a constant unbalanced torque is to 
produce a uniformly accelerated rotation in a mass, and hence that relations 
hold between a, co and 6 identical with those which obtain in the case of 
uniformly accelerated translatory motion. 

Denote the angular velocity of such a body by co, its angular acceleration 
by a, and the angle described in time t under the action of the constant 
torque by 6. Then 

dd dco d'^O 

—r- = CO, -7— = -7— = a. 

dt ' dt dp 

From this it follows that 



CO = at, 6 = \(xt^j (o = \/2a6, 
which formulae are of the same character as those already proved for uni- 
formly accelerated translatory motion; viz.: — 

V = at, s = iat^, V = \/2as- 

It will furthermore be seen that a simple relation exists between the work 
done upon a rotating mass by a torque T and the resulting kinetic energy; 
i. e., TO = ico^I. For calling F, r, respectively, the force and arm of the 
torque, we have Td = FrO = FS = \oj^I. (See § 46, p. 21.) 

It will be noted that the moment of inertia, /, takes the same place in the 
dynamics of rotation that the mass M takes in the dynamics of translation. 
This will be apparent from the comparisons in the following table: — 

Rotation 

w = at 
6 = ial' 
CO = \/2ad 
T = la 
Tt = Ico 
K = iico' 
TO = ^Ico' 

90. Ballistic Pendulum. — A heavy block of wood is suspended on 
knife-edges. A Ijullct whose velocity is to be measured, is fired into it, 
in a line at right angles to the axis of suspension, producing an angular 



Translation 


V = 


.at 


s = 


iat^ 


V = 


\/2 as 


F = 


-- Ma 


Ft = 


= MV 


K = 


= ^MV^ 


FS 


= ^MV^ 



ANGULAR MOMENTUM. 39 

velocity cu. The deflection of the pendulum is measured and w determined 
from this. Calling m the mass of the bullet, v its velocity, / the moment 
of inertia of the pendulum, k the distance from the axis of suspension to 

the line of entrance of the bullet, loj = mv k and v = — ^ . In strictness the 

mk. 

mass of the bullet should be taken into account in the value of /, but this 
is so small that practically it is unimportant. 

In order not to jar the support of the pendulum the line of fire should 
pass through its center of percussion as will be explained later. 

The moment of inertia may either be calculated or determined experi- 
mentally. 

91. Moment of Momentum. Angular Momentum. — The moment 
of a force F relatively to any point O is Fro where ro is the normal drawn 
from O to the line of action of F. In like manner if a mass moves with 
a velocity v, its moment of momentum relatively to O is To = mvro, Vo being 
normal to the direction of v. This quantity, also called angular momentum, 
represents the rotary effect produced by the momentum mv. 

92. Conservation of Angular Momentum. — The angular momen- 
tum of a system of bodies is not in any way altered by the mutual 
actions of the masses composing that system. 

Let Wj W2, Fig. 18, be two masses, and let their angular momenta be 
taken relatively to an axis passing through any point, as O, normal to the 
plane of the paper. The action of 
Wi, ^2 on each other will be along the m/ 
line joining them. As action and re- 
action are equal and opposite the force 
F^ with which m^ is drawn toward niz 
will equal the force F2 by which m^ is 
drawn toward Wj. Under the action of this mutual attraction there 
will be generated in each in time t a quantity of momentum such that 
Ft = m^Vj_ = m^v^. But as will be seen from the figure m^v^ro is the angu- 
lar momentum of Wi relatively to O and Wg'z^a^o that of W2 relatively to 
the same point. Since these are in opposite directions the resultant angu- 
lar momentum = o. Hence they can in no way alter the previously ex- 
isting angular momentum of the system. 

What is true of two masses Wj, W2 is equally true of any number, for the 
effect of any one mass upon any other in the system can be dealt with in 
the same manner, and since the effect of every individual mass upon every 
other is to produce equal and opposite angular momenta with therefore a 
resultant of zero, the total mutual effect of the masses composing the system 
must also be zero. 

93. Conservation of Areas. — It follows from the preceding demon- 
stration that if we suppose a rotating mass Wj with radius r^ and velocity 
Vj so to move that its radius changes to ^2 the velocity will become 7^2, having 
a value such that m-sV^r^ = Wj'L'an. Hence considering a very brief time dt^ 
we have r^Vj^dt = ^27^2^/. But the first member of this equation is double 
the area of the elementary triangle described in time dt by the mass when 
its radius is r^ and the second double the area described by it when its 
radius has become rz- Hence the areas traversed by the radii vectores in 
equal times are the same. 




40 



MECHANICS. 



Because of this fact the conservation of angular momentum has some- 
times been called the " conservation of areas." Kepler's Second Law is an 
illustration of it. 

It will be seen from what has been said that if a portion of a revolving 
mass is transferred from a point near the axis of revolution to one farther 
removed from it the angular momentum of the mass is diminished by the 
same amount as that by which the angular momentum of the portion moved 
is increased. And while the movable part is gaining speed it reacts on the 
mass as a whole. 




uj^y sin BAC^ we have 



X 

y 



(i). 



COi 



94. Composition of Rotations. — If the angular velocities of two 
rotations are represented by the two adjacent sides of a parallelogram the 
diagonal of that parallelogram will represent the angular velocity of the 
resultant. 

Let AB, AC (Fig. 19) represent component angular velocities cOi, a;,. 
Consider any point P whose coordinates referred to AC, AB are x,y. 

Draw PM, PN normal to AB, AC. 
The downward linear velocity of P 
due to the rotation AB is cOiPM = 
oji X sin BAC. The upward lin- 
ear velocity of P due to the ro- 
tation ^C is (JJ2PN = Wij sin BAC. 
For points on the resultant axis 
these opposite rotations must be 
equal. Placing cOiX sin BAC = 

AC 

= — , which is the equation of the di- 
AB' ^ 

agonal AD. This diagonal therefore represents the resultant axis in 

direction. 

AD furthermore represents the resultant rotation in magnitude. For, 

consider the motion of the point C. Its linear velocity due to the resultant 

rotation must be equal to that due to the simul- 
taneous action of the two components. But the 
velocity due to rotation AC = o. That due to 
AB = AC sin BAC X ^^i- That due lo AD = 
AC sin DAC X cor, calling cor tlic resultant an- 
gular velocity. Hence as these last nul^t l)e ecjual, 
AC sin BAC X co^ = AC sin DAC X cor or 

sin BAC , „ sin BAC 

(Or = CO. = AB — = AD. 

sin D/IC sin DAC 

95. Gyroscope or Gyrostat. — Referring to 
Fig. 20, \v{ ])C, the axis of spin of [\\c disk, be 
designated 1)\ A', the axis I-I\ In T, the vertical 
;i\is //(/' l)\- /. Assume that there is i)erf(>ct free- 
dom (jf motion about each of these axes. With the 

I Hi JO. 




GYROSCOPE. 



41 




Fig. 21. 



instrument balanced so that its center of mass lies at the center of the ro- 
tating disc AB, the axis of spin remains parallel to itself in whatever 
manner the instrument as a whole is moved. If, however, the instru- 
ment is unbalanced, as e. g., by hanging a weight from D, a precessional 
rotation takes place about Z, instead of rotation about Y such as would 
occur under Kke circumstances were the disc at rest. 

The precessional motion results from the combination of two rotations, 
the spin of the disc about X and the rotation about Y due to the couple 
produced by the suspended weight. 
Assuming the direction of spin to be 
right handed as seen from D we may 
represent the angular velocity about 
DC by Ox, Fig. 21. 

The angular velocity due to the 
couple produced by the suspended 
weight may be represented by Oy. 
Then zO = Or, will represent the re- 
sultant couple, in magnitude and di- 
rection-. The axis of spin will there- 
fore move in such manner as to tend to approach Or, as indicated by the 
arrow m X O Y, remaining, however, in a horizontal plane, rotating about 
O Z, Fig. 21, or H G, Fig. 20. But as this rotation carries the axis F E 
with it, the direction of Oy and hence of Or is constantly changing, per- 
forming a rotation about O Z\ whence the precessional movement is con- 
tinuous. 

It is easily seen from the construction of Fig. 21 that if either the direction 
of spin about DC or that of the gravitational couple about FE is reversed 
the direction of precession will Hkewise be reversed. 

The following is a general explanation on dynamical principles of the 
phenomena just considered. Fig. 22 represents a disc rotating right- 
handedly about an axis through O 
perpendicular to the plane of the paper 
as an. axis of spin. Suppose that a 
second rotation is given to the disc, 
about YOY, the upper half moving 
toward the eye. The particles in 
quadrant 2 on account of the spin 
about O are moving upwards and in- 
creasing their distance from YOY. 
Hence on account of their momentum 
their effect is to produce a tendency 
in 2 to move away from the observer. 




42 MECHANICS. 

In quadrant i on the other hand the particles are descending towards 
YOY, and on account of their momentum the diminution in their radii of 
revolution about YOY will cause a pressure throughout i towards the eye. 
In like manner in quadrant 4 there will be a pressure toward the eye and 
in quadrant 3 away from it. Hence a rotation will be produced about 
ZOZ as an axis. 

The action described above is analogous to the case of a ball whirled 
by a string, which when the string is lengthened so that its radius of revo- 
lution is increased has its velocity correspondingly slackened, and which 
if it were being pushed forward would press backward in opposition. The 
conception of the action may also be helped by supposing the disc to be a 
rotating disc of liquid contained in a flat box, mounted like the gyroscope. 

The several actions take place in accordance with the principle of the 
conservation of angular momenturr. (See § 92, p. 39.) 

The great resistance opposed by a gyroscope to sudden angular devia- 
tion of its axis of rotation is due to the action just explained. The com- 
bination of couples produces a resultant couple such as would generate a 
precessional motion, and if the impressed force is such as to prevent this, 
the resisting effect due to the momentum of the wheel becomes very great. 

Other gyroscopic phenomena are to be explained according to the prin- 
ciples just laid down. 

Thus with diminishing speed the precession is faster because Ox becomes 
less in proportion to Oy. If the weight on D is increased or if this point is 
pushed down the rate of precession is increased; if pushed up, diminished. 
If the axis of spin is tilted the precession is faster as the effective value of 
Ox is less. If the precession is accelerated by external force, the center of 
mass of the system rises; if the precession is retarded, the center of mass 
descends. The explanation of this may be seen by combining a rotation 
about the axis OZ with Or. If the axis HG is fixed, so as absolutely to 
prevent precession, the added weight will produce rotation about FE, as 
if the disc were not rotating. There will be a torsional stress, however, in 
the vertical axis due to the precessional tendency. Retardation of pre- 
cessional motion by friction of the axis HG will cause a gradual depression 
of the axis of spin. 

96. Phenomena and Applications of Gyroscope,— Rolling hoop; 
bicycle; game of diabolo; rifled guns; steadying g}TOscope on vessel to 
prevent roUing; steadying gyroscope and steering apparatus in Howell 
torpedo; Obry's steering apparatus in Whitehead torpedo; Brennan's 
mono-rail system. Also various devices proposed for determining latitude 
at sea; for transferring directional line, as, e. g., the meridian, from surface 
to bottom of mmc, etc., but none of them as yet practical. 

97. Astronomical Gyroscopic Phenomena. — Parallelism of axis of 
earth to itself, causing phenomena of seasons. Precession of the equinoxes, 



PERIODIC MOTION. 43 

caused by action of sun on equatorial protuberance of earth, with period 
of about 26,000 years. Nutation, caused by action of moon on equatorial 
protuberance, with period of about nineteen years. Periodic variations of 
latitude investigated by Chandler, cause uncertain. 

Application of principle of gyrostat by Kelvin to explain hypothesis of 
elastic-solid luminiferous ether. Kelvin's hypothetical gyrostatic elastic 
atom. 

98, Top, — The various precessional phenomena of the top are to be 
explained upon the same principles. The rise of a top from an inclined 
to a vertical position is due to the combination of precessional motion with 
the rolling motion of the peg. The roll tends to carry the top along faster 
than the precession and so to accelerate this, thereby causing the center of 
mass of the top to rise. 

Maxwell's dynamical Top. Newton's geometric Top. Griffin's stone- 
pulverizer. 

PERIODIC MOTION. 

99. Definitions. — A particle possesses a periodic motion when it 
successively traverses the same path, returning to any given point after 
having passed through a complete cycle of changes. Such motion may 
take place in a closed curve, an arc, or a straight line. The time occupied 
in completing one cycle is called the period. 

The earth revolves about the sun in an elliptical orbit, with a period of 
one year; a "seconds" clock-pendulum vibrates in a circular arc, with a 
period of two seconds; a body suspended by an elastic cord executes a 
vibrating, periodic motion in a vertical line, with a period equal to the time 
occupied by one complete "up and down" motion. 

Any periodic motion can be represented by a curve constructed with 
times as abscissas and the corresponding displacements of the particle as 
ordinates. Such a curve is evidently periodic. When thus constructed 
it is a curve of spaces or displacements. 

A curve representing periodic motion may also be constructed in which 
times are abscissas and the corresponding velocities are ordinates. This 
is a curve of velocities. 

The curve of velocities can always be derived from the curve of spaces, 

and conversely, since v = -y— 

at 

100. Simple Harmonic Motion, — An extremely important case of 
periodic vibratory motion is that known as simple harmonic motion. 

Referring to Fig. 23, let us suppose a particle P to revolve uniformly in 
the circle there shown, whose radius is r, and hkewise that a second particle 



44 



MECHANICS. 



Pi moves to and fro along the vertical diameter of that 
circle in such manner as always to coincide in position 
with the projection of P on that diameter. The 
motion of the vibrating particle is a simple harmonic 
one. 

The position in its path of the vibrating particle at 
any instant is denoted by what is called the phase of 
the vibration. This term is defined more precisely as 
follows: "The phase of a simple harmonic motion at 
any instant is the fraction of the whole period which has elapsed since the 
moving point last passed through its middle position in the positive direc- 
tion." The phase is measured by the angle in Fig. 23. 

When a particle after displacement executes a vibration of this character 
the restoring force at each point must he proportional to the displacement. 




Fig. 23. 



The acceleration at any point P^ is given by the equation 



d^y 

dF 



= a. 



But 



y = r sin = r sm cot, caUing oj the angular velocity and / the time corre- 
sponding to the phase-angle 0; that is, the time occupied by the vibrating 



particle in passing from O to P^ 



1 hen - ,. = ojr cos cut and 



dt dt^ 

— ofr sin cot = — ufy\ whence acxy, the displacement of the particle; 
and the same must be true of the restoring force, since F = Ma. The 
opposite sign of a and y indicates that the acceleration is opposite in direc- 
tion to the displacement. 

The converse of the proposition is likewise true, since this law of force 
can produce only one mode of variation of the acceleration. 

If T be the time of one complete revolution of P and hence also the time 

27r 271 

of one complete vibration of P,, T = — - and oj = ^=-- 

Cx) 1 

It has just been shown that a = ofy (dropping the minus sign as indi- 



cative only of direction) whence co^ = 



T = 271 



V 



y _ 



= 27: 



V 



(r)'"" 



'displacement 
acceleration 



It also appears from this expression that simple harmonic vibrations arc 
isochronous, since the ratio of y to a in any particular case is a constant 
and inde])cndent of r. 

loi. Equations of Harmonic Curves. — Ii is .ipjKinMit that the 
movement of a particle ])()ssessing a simple haniionir motion is repre- 

277 

sented by tiie (-(luiition y -^ r sin 0, or y -^ r sin col, where co -- - y being 



PERIODIC MOTION. 45 

the displacement. Hence, denoting times by abscissas, the equation of the 
curve of displacement is y = r sin cux, which is the equation of a sinusoid, 
r being the amplitude of vibration. 

ds 

The corresponding curve of velocities is readily found. Since v = — ^ 

and in this case s = y = r sin cut, v = cor cos cot. Hence, taking velocities 
as ordinates and times as abscissas, the equation of this curve is 
y = ojr cos wx. This is also a sinusoid, but displaced along the axis of X 

it 
relatively to the former curve by a distance — corresponding to a quarter 

of a cycle or vibration period. 

Apart from the demonstration it is easily seen that the maximum velocity 
of the vibrating particle corresponds to displacement = o, while the maxi- 
mum displacement corresponds to velocity = o. 

The equation y = r sin cot assumes that for ^^ = o, / = o, i. e., the particle 
is assumed to be at the middle of its path at the beginning of the time in- 
terval considered. Hence the curve representing the motion passes through 
the origin. If when / = o the particle occupies any other position, the 
phase being for example an angle d in advance of the zero position, the 
equation may be written /y = r sin {cot -\- d) or y ^ r sin {cox -\- d), which 
is a more general form than the preceding, d is called the angle of epoch. 

Fig. 24 illustrates the construction of a curve representing a simple 
harmonic vibration. The 

abscissas are made pro- ^,,<'--^->>.- -y""^ y — \ 

portional to times, and f / \ / i \ / \ 

hence to arcs on the refer- r o J ' ' ' ' ' \ ■ ' } ' ' / 

ence circle, i. e., to the \ / >v ' / 

angle 0. The ordinates ' 

are made proportional to 
displacements parallel to F, i. e., to sin d. 

In the study of vibrations it is often more convenient to measure the phase- 
angle from the axis of Y rather than from X, and to use cosine rather than 
sine functions. In this case a vibration parallel to Y is represented by the 
equation y = r cos {cot + d) and the corresponding harmonic curve by the 
equation y = r cos {cox + d). 

102, Permanent Record of Vibrations.— Trace by pencil carried 
at end of elastic vibrating bar against sheet of paper, moved uniformly at 
right angles to direction of vibration. Curves drawn on smoked glass by 
style attached to uniformly-moved vibrating tuning-fork, or on smoked glass 
carried by vibrating tuning-fork, the style being moved uniformly in a 
straight line. 



Fig. 24. 



46 MECHANICS. 

103. Equation of Wave Motion in Elastic Medium. — From 
the equation of the harmonic curve we can derive the equation of a simple 
wave propagated in an elastic medium, as, for example, the sound-wave. 

From the principles of wave motion it follows that the instantaneous value 
of y at time / for any point of the wave whose abscissa is x is the same as 
the value which it had ^\ x = o (the source of the wave disturbance) at a 
previous time earlier than that under consideration by the interval which 
has elapsed between the starting of the wave from x = o and its arrival at x. 

Denoting by T", as before, the time of one complete vibration and by m 
the number of cycles (in general not an integer) that have elapsed since the 
wave started from the origin, this interval = mT. Hence 

27r / 
y = r ^mcD {t — mT) = r sin ^ (t — mT) = r sin 271 (^ m). 

If we denote by X the length of the wave under consideration, mX = x 

and m = -^, since m waves have been generated while the disturbance was 

travelling from :x: = o to x. Hence y = r sin 271 {-=r V), which is 

■L A 

the form in which the equation of the elastic wave is usually given. 

As the velocity of propagation of the wave is 1^ = 71X = —^w, {n = fre- 
quency) this equation may also be written in the form 

t X 

>' = r sin 27r (-^ — -7=). 
T vT 

It will be seen on consideration that the equation of the wave is that of 
the curve shown in Fig. 24. The axis of Y at the instant under considera- 

27r/ 
tion, /, for which the phase \s = — — , must pass through the point for 

which the ordinate \s y = r sin 27z -=^. This defines the location on X of 
:>; = o at that instant. 

104. Combination of Parallel Vibrations and of Harmonic 
Curves, — It is evident that for each value of x on the resultant curve 
the ordinate will be equal to the algebraic sum of the ordinates of the com- 
ponent curves. Hence if the equations of these are )'i = r^ sin {lo^x + di) 
and y^ — r^ sin ((t>2^ + ^2) the equation of the resultant curve will be 
y = Vj^ sin {(jj^x -f- oJ -\- r2 sin {uj^x + o^). 

This equation represents a periodic curve, since if co^x, CO2X both be 
increased by 271 or W27r, the values of y recur. The period of the resultant 
curve depends upon the ratio of co^ to oj^. If this reduced to its lowest 
term is p : q the resultant period will be the time taken to make p vibra- 
tions of the first or q vibrations of the second com]X)nent. 

In general the nature of the results is better a])prchended from a study 
of the grnphical combination of such curves than from analysis. 

Various methods have been described for drawing such compound har- 
monic curves mechanically. 



COMPOSITION OF VIBRATIONS. 47 

105. Applications, — Curves drawn with tuning-forks, the smoked 
plate carried by one fork, the style by another. Beat Curves. 

Mechanical synthesis of harmonic curves. Curve-tracing machines of 
Donkin and Cady. Kelvin's Tide-Predicting Machine. Synthesis and 
analysis by Harmonic Analyzer of Michelson. 

106. Composition of Parallel Vibrations of Same Period. — 
A case of the composition of simple harmonic curves which is of great im- 
portance because of its application in theoretical optics is that in which the 
periods of the components are the same. Under these circumstances the 
resultant is a simple harmonic curve having the same period as that of the 
components. This will appear from the following consideration. 

Writing i02 = coi the equation of the resultant curve becomes 

y = fj^ sin (cjjX + dj) -{• r^ sin {co^x + ^2) 

whence by expanding the values of sin {oJj^x + <5i), sin {oj^x + ^2) we derive 

the equation 

y = (^i cos <5i -f- r2 cos ^2) sin ojj^x -f (fi sin ^i -f Vz sin ^2) cos ojj^x. 

In this equation we may write r^ cos (^j = r^^ cos ^i + ^2 cos ^2 (i) and 
r^ sin ^3 = fi sin ^i + /'2 sin ^2 (2), subject to a subsequent determination 
of the values of ^3 and 03. Then 

y = r^ cos <^3 sin oj^x -\- r^ sin d^ cos oJj^x = r^ (cos d^ sin co^x + sin d^ cos ojix) 
ory = Ts sin {co^x + 03) 

This represents a simple harmonic curve of period oj^, amplitude r^, and 
epoch d^. 

The values of r^ and d^ are easily found as follows: — 

By squaring equations (i), (2), adding the results and simpHfying we 
have ^3^ = r^s r^^ ^ 2r^r2 cos (^i — ^2), 
whence the amplitude of the resultant is 

^3 = V^i^ + r^^ + 2r^r2 cos (^i — ^2). 

-p.. ... , . , , . ' ^ ^ ^i sin Oj + ^2 sin ^2 

Dividmg (2) by (i.) we have tan 03 = 



^i cos ^i + ^2 cos ^2" 

The cases where the amplitudes fi, fj are equal should be particularly 
noticed. If in addition to this equality d^ — dz = o, r^ = 2f i ; that is, if 
the components are in the same phase the amplitude of the resultant curve 
is double that of either component, li d^ — dz = n, r^ = o; that is, if 
the components are in precisely opposite phases, the amplitude of the re- 
sultant = o and the curve is reduced to a right line coinciding with the axis 
of X. For intermediate values of the phase-difference the amplitude varies 
between these two extremes. 

It follows furthermore from what precedes that the addition of any 
number of simple harmonic curves of the same period gives as a resultant 
a simple harmonic curve also of the same period. For the components 
may be combined successively, each with the resultant of those previously 
taken. 

107. Combination of Rectangular Harmonic Vibrations. — ■ 
Measuring the phase-angles from Y and X respectively, the component 
vibration parallel to Y may be represented by the equation y = r cos cot, 
that parallel to X by the equation x = r^^ cos (ajj + d). 

For purposes of illustration it will suffice to consider the simplest case, 



4^ MECHANICS. 

that in which the periods are the same and hence w^ = co. We have, there- 
fore, X = ^i cos cos col — ^i sin 3 sin cut. But cos cot = — , sin cot = 



\/i — i. Hence :)t; = r^ cos o (^~J — r^ sin o ( v/i — ^\ and 

^^ + r^^ cos^ ( "^ ) - 2 r^jc cos ^ ( ^'^~ ) ^ ^^^ ^^^^^ ~ "^i' ^^"''^ ( "^ ) ' 
which may be written 



X'' y"" 2xy ^ 

H cos = sm^ 0. 



This is the equation of an elHpsc. 



which is the equation of a straight hne passing through the origin and making 
an angle tan -^ I — j with X. 

X"^ /w2 

If ^ = J (2;r), — + -^ = I, the equation of an eUipse with its axes 

lying in X and Y. If in addition r = r^, x^ + y^ = r^, and the ellipse be- 
comes a circle. 

X y 

li d = ^ (271), -^^ = -~, the equation of a straight line through 

^i r 

the origin and making an angle tan "M ~ / ) ^^'i^h X. 

If ^ = } (27r), 1- -^- = I, and if also r - }\, x' + y^ = r^. 

The ellipses represented by the preceding equations are always inscribed 
in a rectangle of which 2r, 2r^ arc the sides. The points of contact of the 
ellipse with the sides of this rectangle may be found by determining the 
intersections of the curve with the lines jc = ± ;-, y = ± )\. 

If the periods of the vibrations to be combined have any other ratio, 
e. g., I : 2, or I : w, in the general equation we must make co = 2co^ or 
(o = noji. But except in a few special cases the resulting curves are of 
very complex character and are more readily studied ])y gra])hical methods. 
(See text-book.) 
If the dilTerence of phase represented by varies graduall}- from o to 2- 

Ihe curve will gradually ])ass in succession tlirough 
all its corresponding variations. Fig. 26 illustrates 
these for the several ratios there indicated. 

108. Applications. — I^lackburn's Pendulum 
(I'ig. 25); llarniouograph Pendulums; Wheat- 
stone's Kalei{lo])lu)ne; Lissajous' Curves (I'^'g. 26); 
Ivissajous' Com])arator. Mechanical C'ombina- 
i'ui. 25. tion by Machines of Picki^ring, Donkin, and others. 




rOURIER S THEOREM. 



49 



I : I 






I : 2 













Fig. 26. 



109. Fourier's Theorem. — "If any arbitrary periodic curve be 
drawn, having a given wave-length X, the same curve may always be pro- 
duced by compounding harmonic curves (in general infinite in number, 
having the same axis, and having X, ^X, JA, . . . for their wave-lengths. 

" The only limitations to the irregularity of the arbitrary curve are, first, 
that the ordinate must be always finite; and, secondly, that the projection 
on the axis of a point moving so as to describe the curve must move always 
in the same direction. 

" These conditions being satisfied, a wave of the curve may have any 
form whatever, including any number of straight portions. 

" Anahtically the theorem may be expressed as follows: — 

"It is possible to determine the constants C, Cj, C2, etc., a^, a2, etc., so 
that a wave of the periodic curve defined by the equation 



^^ = C -I- C, sin (^^ + a,^ + C.sin ( 



27:x \ 

2^- + a,) + 



^ = 00 



or 



y = C + I Ci sin r — ^ -1- an 



shall have any proposed form, subject to the conditions mentioned above." 
The preceding statement is extracted from Donkin's "Acoustics" (Clar- 
endon Press Series). 



50 MECHANICS. 

1 10. Vibrations of Elastic Bodies.— Since in the case of an elastic 
body subjected to any kind of deformation the restoring force within 
certain limits is proportional to the displacement, it follows that vibra- 
tions of small ampHtude performed under the influence of elasticity are 
simple harmonic in character, and isochronous. 

In case the amplitude of the vibration is so great that the restoring force 
is not at each point proportional to the first power of the displacement, 
the motion while still periodic is no longer simple harmonic in its character. 
It can be shown by analysis that under these circumstances the vibration 
is a compound harmonic one, represented by the resultant of a series of 
simple harmonic vibrations of frequencies n^ 2n, 3W, etc., the particular 
harmonics present being determined by the law of variation of the restoring 
force. 

Experimental evidence of these effects is clearly seen in the case of a 
vibrating tuning-fork. When gently bowed only the fundamental (fre- 
quency = n) can be heard, but with strong bowing the octave (frequency 
= 2n), the fifth of the octave (frequency = 3^), and even higher harmonics 
also appear. 

A like occurrence, to be explained in a similar manner, is the occasional 
appearance of harmonic notes in telephonic transmission when a micro- 
phone transmitter is used. 

The mathematical principles underlying these several phenomena are 
stated in § 109, p. 49. 

III. Acoustic Vibrations. — The formula T = 271^ J-... = 211 yl A^ 

(in which d is the displacement) enables us to obtain the laws of vibration 
of bodies vibrating under the influence of elasticity. 

The following demonstrations are of importance in acoustics, as giving 
the laws of vibration of sounding bodies. 

I . I j"^ 

Let n = -„ be the frequency of vibration. Then n = — a/ — = 



2W 



where F is the restoring force and M the mass in vibration. 



27r "^ Md 

For a bar or wire, M = length X cross-section X density = I s w, and 



therefore n cc \l 

Iswd 



Esd 
For a bar or wire executing longitudinal vibrations, F = — - — , E being 

t 



Young's modulus of elasticity; whence w oc^/ _:! — oc , \/ 



E 

w 

Hence for longitudinal vibrations the frequency is independent of the 
section of the bar, and varies inversely as the length, inversely as the square 
root of the density and directly as the square root of the modulus of elas- 
ticity of its material. 

I"\)r an elastic string vibrating transversely, calling P the tension of the 
string when deflected, F \ P w 2d \\ I. (See Fig. 27.) 



PENDULUM. 



51 



Hence i^ = ^ and. oc^_l^ oc^V- ^iV- 

as ^ = 7ir^. 

Hence for such a vibrating string the frequency varies inversely as the 

length of the string, directly as the square root of the tension, inversely as 

the diameter and inversely as the square root of the density. 

Abk^d 
For a vibrating elastic rod of length /, breadth b, depth k, F = 



bk^ 



(X 



■fV 



/3 



W 




^ being a constant. Hence w cc \ _ _ a a/ ^ 

That iSj for a bar vibrating transversely the frequency is independent of 
the breadth, inversely as the square of the 
length, directly as the depth, and inversely 
as the square root of the density. 

These laws of variation of frequency are /^^ 

independent of the particular manner in 
which the bar is supported, only the abso- Fig. 27. 

lute frequency being affected by this. 

112. Circular Pendulum. — The time of vibration of a circular pen- 
dulum swinging in an indefinitely small arc 
may be found as follows: — The vibration is 
simple harmonic since the accelerating com- 
ponent is F = W sin d = WO for small 
angles. (Fig. 28.) Let O P = / and P iV = r. 

4 



Then P = 2/ = 27r 



27r 



v 



/ 



and / 



^ a 

7l\- 



l sin 



= 27r 



^sin d 




Fig. 28. 



113. Cycloidal Pendulum. —The law of curvature of the cycloid is 
such that with the cycloidal pendulum the accelerating (tangential) com- 
ponent of the weight of the 
vibrating particle {W sin 6) 
is always proportional to 
the arc of displacement. 
Hence the vibrations are 
isochronous for all arcs. 

This property of the cy- 
cloid follows directly from 
the fact that with it the 
length of the arc included 
between the vertex and 
any point E (Fig. 29) is 



d = 



4> 



j\a cos -^ where 
2 



is the radius of the gen- 




52 MECHANICS. 

crating circle and </> the angle made with the vertical by this radius drawn 
through E. Calling the angle of deviation when the vibrating particle 

is at £, -^ = (90° — 0). The restoring force at E is i^ = W sin = 
2 

W cos (90° — 0) = W cos -^. Hence F oc cos -^ a d. 

2 2 

It will also be seen from the demonstration in § 112 that the time of 
vibration for all arcs is / = 71^/ 

114. Damped Vibrations. — When a vibrating body executes free 
vibrations the amplitude of these gradually diminishes owing to the energ}^ 
expended in overcoming external or internal resistances. This effect is 
called damping. 

In general the effect of the resistance is proportional to the velocity 
simply, and the diminution of amplitude is proportional to the amplitude 
itself at each instant. That is, denoting amplitudes by y and times by x, 

-Z = — Jiy whence — - = — k dx. Hence by integration 
dx y 

log y = — kx + constant and y = ce~^-'^. 

It follows from this equation that if we designate by ym and y„ respectively 
the amplitudes of the decreasing vibration after corresponding times :V;„ , .y„, 
and if during the interval Xn — Xm , N vibrations have been made, then 

^'^^ ^-^ =A, a constant, which is known as the loparitlimic decrement 

N 

of the vibration, and is evidently the decrement per vibration of the logarithm 
of the amplitude. 

It will also be seen that the vibration of a particle subject to damping 
may be represented graphically by replacing the reference circle in Fig. 
24 by a logarithmic spiral. 

The periodic curve representing a damped vibration must be character- 
ized by a diminution of its ordinates from those of the sine curve repre- 
senting the corresponding displacement with an undamped vibration of 
the same period, which follows the law represented by the equations just 
demonstrated. Hence the equation of such a harmonic curve will take 
the form y = e~^^ r sin {cox -|- <?). 

Fig. 30 illustrates the method of constructing the curve representing 

a damped sim])le har- 
monic vibration. The 
\\ abscissas are made ])ro- 

i ! \ / N. ^ — \ ]H)rtic)nal to times and 

" ' hence to the angle 0. 

The ordinates are made 

,, proportional to the value 

I"i(;. 30. j; I 

for tlie angle of the 
ordinate of the extremity of \\\v radius vector of the logarithmic sj)iral 
of reference, /. e., to r sin 0. 




FREE AND FORCED VIBRATIONS. ■ 53 

115. Stroboscopic Method of Studying Periodic Motion. — 

When a rapidly moving body is illuminated by a flash of light of such brief 
duration that the body does not traverse a perceptible space during its 
continuance, the object appears to be at rest. This is the case, for ex- 
ample, with a swiftly revolving wheel illuminated by the electric spark. 
If there is a series of brief illuminations the object is seen in a correspond- 
ing series of positions. If a body possessing a periodic motion is illumi- 
nated by a series of recurrent flashes, and if the interval between each 
flash is such as to coincide exactly with that period (or with n times the 
period) the body will appear to be absolutely at rest, however great its 
velocity may be. It will be seen continuously owing to the persistence of 
vision. If the interval between the flashes is slightly less than the periodic 
time the body will apparently possess a slow retrograde motion. If the 
interval is slightly greater than the periodic time, there will be an ap- 
parent slow forward motion of the body. In this way changes of con- 
figuration or of form may be studied. 

The flashes may be obtained by any convenient method, as by the 
electric spark or by throwing the light from an electric arc through a rota- 
ting disc with radial slits; or the moving object while continuously illumi- 
nated may be viewed directly by the eye, looking through a rotating slotted 
disc. 

116. Illustrations and Applications. — (a) Single flash. Flash- 
light photography. Rapid-shutter photography. Study of trotting horses, 
etc., by Muybridge. Spark-photography of flying bullets by Mach and 
Boys. 

(b) Stroboscope. — Determination of period of rotating wheel. Study of 
motion of chain of small bones in human ear by Mach. Determination of 
frequency of tuning-fork. Stroboscopic study of alternating arc in Rogers 
Laboratory, M. I. T. Use of microstroboscope in telephonic studies in 
Rogers Laboratory. 

Toys; e. g., Phenakisticope of Plateau. Stroboscopic discs of Stampfer. 
Dedaleum of Horner. 

(c) Kinetoscope; Kinematograph. Early application of photography 
by Muybridge to study of gait and postures of animals. Representation of 
movement of express train, of mountain ascents, etc. Representation of 
progressive motion of waves by Miiller and Wood. Study of plant growth. 

117. Free and Forced Vibrations. — If a body so conditioned as 
to be capable of entering into a state of vibration is subjected to the action 
of a disturbing impulse and then left free, it will execute what are called 
free vibrations, the period of these being determined by the mass, form, 
and mechanical conditions of the body. If such a body is subjected to the 
continued action of a periodically varying force, external to itself, this will 
act to produce in the body vibrations of the same period as that of the force 
variation. These are called forced vibrations. 

If the varying force coincides in period with any vibration which it is 
possible for the body to execute, extremely strong forced vibrations may 
be produced, even though the efl'ect of each separate periodic action of the 



54 MECHANICS. 

force Is snicall. In this case vibrations thus excited are often called sym- 
pathetic vibrations or syntonic vibrations. 

When the action of the force is reciprocating, so that periodic reversals 
in its direction occur, it can only produce strong sympathetic vibrations 
in a body which is capable of vibrating in the same period. But if the force 
acts intermittently, producing a series of equally timed impulses, it may 
excite sympathetic vibrations in a body which is capable of vibration in 
any period which is an integral number of times that of the impulses. 

Illustrations of the former of these conditions are found in the acoustic 
phenomena of the sympathetic tuning-forks and resonance. 

The latter conditions are found in many cases of forced mechanical 
vibrations, as in the ringing of a church bell, in Galileo's experiment in 
which a heavy pendulum is set into vibration by properly timed puffs of 
the breath, in the vibrations of suspension and other bridges excited by 
marching bodies of troops, and in the vibration of mill buildings by 
machinery. 

Ii8. Applications. — Hardy's- "Noddy" used in pendulum experi- 
ments. Frahm's Resonance Apparatus. Hartmann and Braun's Fre- 
quency Meter for Alternating Currents. Helmholtz's Harmonic Tuning- 
forks. Clocks governed by action of electro-magnet upon Pendulum. 



119. Phase Relation. — There is always a certain difference of phase 
between the force which acts to produce the forced vibration and the re- 
sulting forced vibration itself, the latter lagging behind the former. The 
amount of this difference depends upon the relation between the periods 
of the force and the free vibration. If these are precisely the same, the case 
of perfect unison, the phase-difference can be shown to be \ 7:, that is, 
one-fourth of a period. If the natural frequency is the greater, the lag of 
the forced vibration is less than a quarter period; if the natural frequency 
is the lesser, the lag is between one-quarter and one-half period. 

Illustration: Phenomena observed with Sympathetic Pendulums. 

120. Coexistence of Free and Forced Vibrations. — When a body 
is set into forced vibration by the action upon it of a periodic force 
there is often noticed at the beginning of the operation, before the forced 
vibration is fully established, a rhythmic variation in the amplitude of the 
vibration actually performed. In the case of an acoustic vibration " beats" 
are heard. 

This phenomenon is caused by the simultaneous existence of a free 
vibration, due directly to the disturbance of the vibratory body and the 
forced vibration impressed upon the body. If these two periods are not 
exactly alike beating must occur. 

Illustrations: Beats of tuning-fork electrically driven by a governing 
fork of same frequency. Beats of electrically excited musical string. 
Frahm's Resonance Top. 



INTERFERENCE OF WAVES. 55 

121. Resonance. — In acoustics the term resonance is usually applied 
to the phenomena of forced vibrations, especially when they are produced 
in a mass of air. 

The small density of air enables it to enter into a state of forced vibration 
very readily even when the period of this is considerably removed from the 
period of free vibration, although the resonance rises rapidly in amount 
as the period of the forced vibration approaches that of the free vibration 
of the mass of air. 

Any mass of air or other body which is capable of entering into a state 
of vibration will respond by resonance to sounds having the same pitch 
(ind hence the same vibration -frequency) as any which it can emit in 
consequence of its own free vibration. 

Illustrations of Resonance of Mass of Air: Tuned Bottles. 
Helmholtz's resonators. Open and stopped organ pipes. 

In case a body is capable of vibration in several different periods it will 
respond to any one of the several notes of corresponding vibration-fre- 
quency. It will also respond to several such sounds simultaneously. 

Illustrations: Multiple resonance of organ pipes. Simultaneous 
resonance of organ pipe to two or more forks. 

122. Conditions affecting Damping. — The more readily a body 
enters into a state of forced vibration, the more readily are its vibrations 
damped. For example, a mass of air in a resonator can be forced by a 
powerful tuning-fork to vibrate in a period far removed from its natural 
rate of free vibration ; but the sound of such a vibrating mass of air persists 
only for a very brief period. On the other hand, a tuning-fork can only 
be made to execute forced vibrations when the exciting fork is very closely 
in unison with it; but vibrations excited in any manner in the fork are 
very persistent. 

Whenever a vibrating body causes sympathetic vibrations in a second 
body, the former loses its own energy of vibration more rapidly in proportion 
to the strength of the resonance which it causes. Hence the damping of 
the vibrations of a body in proximity to others capable of vibrating at the 
same rate as itself wall be more rapid than if it is removed from the neigh- 
borhood of such bodies. The reaction of the sympathetically vibrating 
body upon the other is always such as to oppose its motion. 

INTERFERENCE OF WAVES. 

123. Phenomena. — If two trains of parallel waves of equal length 
meet one another and coalesce, the waves constituting the resultant train 
will possess an amplitude equal to the algebraic sum of the amplitudes of 
the components. The wave length will remain unchanged. If the waves 



6 MECHANICS. 



are in similar phase the resultant wave will be of greater amplitude than 
either component, if they are in unlike phase, less. In the particular case 
where the amplitudes of the coalescing waves are equal and their phase 
opposite they will neutralize each other and the amplitude of the resultant 
wave will be zero. This effect is known as interference. If the amplitudes 
of the waves thus compounded are unequal, the interference will be only 
partial. 

This phenomenon is considered analytically in § io6, p. 47 of these 
Notes. 

In most cases of interference the interfering trains of waves start from 
the same source and reacli the place of coalescence by two different paths, 
one of which is half a wave length or an odd number of half wave lengths 
longer than the other. Under these circumstances the two sets of waves 
always meet trough to crest and so interfere, completely or partially, ac- 
cording to their relative ampHtudes. 

In the case of sound waves meeting in opposite phase, i. e., condensation 
to rarefaction, the addition of one sound to another sound of the same pitch 
and loudness, and hence of the same wave length and amplitude, may pro- 
duce silence. In like manner the addition of light to light may produce 
darkness. 

Illustrations: Water Waves. — Absence of tide at Batsha, where the 
tidal wave reaches the port by two channels of different length. Increased 
tide at certain points on eastern coast of England arising from interference 
of wave trains travelling respectively northward from English Channel and 
southward from sea above Scotland. 

Waves in cords. — Stationary waves in Melde's experiment. 

Sound waves. — Herschel's trombone apparatus. Stationary waves 
formed in free air by coalescence of trains of waves, respectivel}' direct and 
reflected from a wall. 

Light waves. — Colors of thin films, as in case of soap bubble. 

GRAVITATION. 

124. Law of Universal Gravitation. — Every particle oj matter 
in the universe attracts every other particle li'ith a jorce varyitii!^ directly as 
the product oj their masses, and inversely as the square oj their distance. 

Hence the mutual gravitation of two masses ?)!, ;;/', at a distance d, is 

^ d' 

The general thcor}- of gravitation, as j)rove(l by XrwliMi during the years 
1666 to 1687, followed from his establishment of tlie following facts: — 



GRAVITATION. 57 

" I . That the force by which the different planets are attracted to the 
sun is in the inverse proportion to the squares of their distances, [i. Re- 
sults from 3d Law of Kepler.] 

"2. That the force by which the same planet is attracted to the sun, in 
different parts of its orbit, is also in the inverse proportion to the square 
of the distance. [2. Results from ist and 2d Laws of Kepler.] 

"3. That the earth also exerts a force on the moon, and this force is 
identical with the force of gravity. 

" 4. That bodies thus act on other bodies besides those which revolve 
around them; thus the sun exerts such a force on the moon and satellites, 
and the planets exert such forces on one another. 

" 5. That this force thus exerted by the general masses of the sun, earth, 
and planets arises from the attraction oj each particle of these masses; 
which attraction follows the above law, and belongs to all matter alike." 
— Whewell. 

The Laws of Kepler referred to above were determined by that astron- 
omer wholly from observation (1609; 1619). They are as follows: — 

1. The planets all revolve in ellipses with the sun in one focus. 

2. The radius vector describes areas proportional to the times. 

3. The squares of the periodic times of the planets are proportional to 
the cubes of their mean distances from the sun. 

125. Value of Gravitation Constant. — The constant ^ in the 
formula for gravitation denotes the attraction existing between two masses 
of one gram each at a distance of one centimeter apart. Its value has been 
determined by the method of Cavendish (1798) in which a small metallic 
sphere carried on the arm of a torsion balance is attracted by a heavy lead 
sphere. The force between the two spheres is known from the torsion 
developed in the wire of the balance. Their masses and distance apart 
are also known, so that the value of ^ can easily be computed. Results 
by Boys, who used a quartz fibre suspension (1895) give as its value 
(j) = 6.6576 X io~^. By comparing the attraction of the lead sphere on the 
sphere of the torsion balance with the attraction of the earth on the same, 
the mean density of the earth can be determined. The value corresponding 
to the value of ^ given above is J = 5.5270. 

Values of </> agreeing closely with that given above have also been obtained 
by other methods. Thus Poynting (1891) measured the attraction between 
two lead spheres by means of a chemical balance of great sensitiveness. 

The law of gravitation appears to hold throughout all distances, varying 
from interplanetary spaces certainly to within a few centimeters and pre- 
sumably until the attracting bodies are in contact. ^ is independent of 
the material of the masses, and of the medium separating them. Also 
gravitation is in no case directive or polar in character. 



58 



MECHANICS. 



126. Diminution of Gravity without Surface of Sphere. 

Newton showed that the resultant effect of all the particles of matter com- 
posing a homogeneous sphere upon a particle outside of it is the same as 
if the total mass of the sphere were concentrated at its center. Hence, 
calling G, Gh, the respective attractions upon a body at the surface of a 
sphere of radius R, and at a distance h above the surface, 



whence 



or 



G :Gh : 



Gh = G 



I I 

~¥ ' {R + hy 
R^ 



{R + hy 
Gh = G ( 1 — y approximately. 



The preceding formula can be applied to the diminution of gravity on 
ascending above the surface of the earth, where G, Gh are the weights of 
the body. Also, it follows that 

gh = g (1 — -^ ) approximately. 



VARIATION OF g WITH ELEVATION. 



San Francisco 

Lick Observatory, Mt. Hamilton , 

Denver 
Pike's Peak 

Kawaihae 

Kalaieha 

Waiau, Summit Mauna Kea 

Do. Reduced to Lat. of Kawaihae 

Tokio 

Fuji, Summit 

Do. Reduced to Lat. Tokio 



Von Jolly (1881) made a direct determination of the diminution of g in 
ascending, by the use of an equal-arm balance with two sets of pans, 
one close to the beam, the other at a distance below it, and consequently 
nearer the surface of the earth. With a distance of 121 metres between 
the pans, a mass of 5 kilograms weighed 32 milligrams more when in the 
lower than when in the upper pan. 

Von Jolly furthermore found that by placing a sphere of lead of mass 



Elevation 
/t. 

375 
4205 


cm. 

979-951 
979.646 


Mendenhall 


5374 
14085 


979-595 
978.940 


Putnam 


8 

6660 

13060 


978.798 
978.485 
978-055 
978.067 


Preston 

a 
(I 



1 2441 


979-84 
978.86 
978.65 


Mendenhall 

11 



EFFECT OF FIGURE OF EARTH. 59 

5775 kilograms immediately under the movable mass when thus lowered, 
the weight of the latter was increased by one-half of a milligram. This 
was due to the gravitational attraction between the two spheres. It follows 
directly from this that the earth's attraction on the movable mass was lo 
miUion times that of the lead sphere, and hence that the mean density of 
the earth should be about 5.7. 

Richarz and Krigar-Menzel using a modification of this method obtained 
a value of J == 5.5, agreeing very closely with that obtained by Boys by 
the torsion balance. 

127. Instruments for Direct Measure of Variations in Gravity. — 
Mass supported by spring; Siemens' bathometer. Threlfall and Pollock's 
quartz- thread gravity balance. Von Sterneck's barymeter; obhque column 
of mercury balanced on knife-edge support. 

128. Effect of Spheroidal Form of Earth. — Analysis and ob- 
servation both show that the force of gravity increases from the equator 
to the poles. The actual gain of weight of a body carried from the former 
to the latter latitude would be about ~^- of its original value. Of this 

approximately -^ arises from the centrifugal force caused by the rotation 

of the earth. The remainder, about -^-, is due to the fact that the earth 

545 

is an oblate spheroid. 

At the poles g = 983.194 cm. 
In latitude 45° g = 980.630 cm. 
At the equator g ^ 978.066 cm. 

Value of g^ for any Place. — If ^o be the value of the acceleration of 
gravity in latitude 0° and at sea-level, the acceleration ^^ in latitude X and 

(2 h \ 
I — -^ V 

This is the value which g-^ would have in mid-air. In the actual case of 
stations in mountainous regions various particular formulae have been used, 
as no general formula can be deduced which will take all local disturbances 
into account. (See § 150, p. 68, for a further consideration of the sub- 
ject.) 

129. Gravity within Sphere. — It can be shown that within a 
homogeneous sphere the weight of a mass would be directly proportional 
to its distance from the center. But this condition is very far from being 
true of the earth owing to the increased density of deeper strata. In fact, 
g Sit first increases with increasing depth. Below a certain but unknown 
depth it must obviously decrease. 

130. Falling Bodies. — Laws. 

I. The acceleration produced by gravity is independent of the mass; 
whence the velocity of a freely falling body is independent of its mass. 



6o 



MECHANICS. 



Experimental Proof. — Cannon-balls of different masses dropped from 
tower. (Galileo, about 1590, at Pisa.) 

Theoretical Proof. — Calling M, M' the masses of two bodies, G, G' 
their attraction to the earth, a, a' the accelerations produced, we have 

G.G' :\M :M', 
but Ma : M'a' ::G:G'; 

whence a = a\ 

II. The acceleration is independent of the material of the body. 
Experimental Proofs. — (a) Falling balls of metal and cork; (b) 

guinea and feather tube; (c) proofs of Newton and Bessel by pendulum, 
to be discussed later. 

III. Velocity is proportional to the time of descent. 

IV. Velocity is proportional to the square root of distance fallen through. 

V. Space described is proportional to the square of the time. 

III., IV., v., follow from the fact that the motion is uniformly accelerated. 
131. Verification of Laws by Experiment. — The three last-men- 
tioned laws of falling bodies may be investigated experimentally by means 
of various devices invented for the study of uniformly variable motion, in- 
cluding that produced by gravity. Of these the following are particularly 
worthy of mention: — 

I. Inclined Plane (Galileo). Apply preceding formula;. (See § 26, 
p. 8.) 

2. Alwood^s Machine (1780) (Fig. 31). The equal 
large masses, M, M are connected by a flexible cord 
running over a pulley A. The addition of a small 
mass m produces a uniformly accelerated motion, in 
which the acceleration is found from the equation 

m 

10= mg = a (2 M + m), whence a = g 

2M + m 

This value of a being substituted in the general 
formulae, we have the velocity acquired by the mass 
and the space traversed by it, which may be com- 
pared with the results of experiment. 

The effect of the pulley A must also be allowed 
for. This may be computed from its moment of 
inertia, but it is best determined by experiment. It 
will be the same as if the masses M, M were increased 

m 



A 



A/CZD 



Fig. 31. 



by a constant amount M' . Making this correction, a = g — 

2 if + M' -f m 

The following demonstration of the value of (/ with Atwood's machine 
is based on the principles of Energy. 



BODY FALLING FROM GREAT HEIGHT. 



6i 



If the heavier mass descends through a distance h and the system ac- 
quires thereby a velocity v, then wh = mgh = \ {2M -{- ni)v^ = i(2ilf + w) 
{2ah), observing that v^ = 2ah, since the motion is uniformly accelerated. 



m 



as before. 




Solving this equation relatively to a, we have ^ = g T,r , j 

3. Barbouze^s Machine. In this machine, a lamp-blacked cylinder is 
attached to the axis of the pulley of an Atwood's machine. Against this 
rubs a style attached to a tuning-fork, which marks equal intervals of time 
by its vibrations, and from the relative length of the sinuosities produced 
with increasing velocity, the laws can be determined experimentally. 

A simpler method, used at the Institute in 1869, consists in causing a 
freely falling glass plate, covered with lamp-black, to press very lightly 
against a style carried by one of the prongs of a vibrating tuning-fork. In 
a later form of the apparatus the fork falls while the glass plate is fixed. 

4. Morin's Machine (Fig. 32). A freely faUing body traces its path 
against a cylinder moving uniformly about a vertical axis. 

By the combination of the two motions a parabola is traced, ^,-4-^ 
showing that the motion of the falling body is uniformly ac- 
celerated. 

In a modified form of the machine, devised by Sir George 
Darwin, the pencil is fixed and the revolving cylinder falls. 

5. Additional Methods. The laws of falling bodies are also 
studied by determining the velocity acquired and space trav- 
ersed, by means of electric chronographs. 

It is evident that any of these devices will enable us to obtain 
a value of g, of approximate correctness. Accurate methods 
of determining this will be explained in the chapter on the 
pendulum. 

132, Case of Body falling from Great Height. — 

Since gravity varies inversely as the square of the distance from. 

the center of the earth, in the case of a body falling through 

a great distance, this must be taken into account. For small 

distances the diminution may be neglected, as at the height 

of a kilometer it is only -^ of the value at the surface. It Fig. 32. 

3188 

can be shown by analysis that the velocity acquired by a body 

falling freely to the earth from an infinite distance would be about 35,000 

feet per second. 

Conversely, a mass projected vertically upward with this velocity would 
not return to the earth. 

This principle has served to disprove an early hypothesis as to the 
lunar volcanic origin of meteorites, and also, by an application of the 
Kinetic Theory to indicate the probable constitution of planetary atmo- 
spheres. 



62 



MECHANICS. 



133. Projectile. — The trajectory of an unresisted projectile is a 
parabola. 

Let O R (Fig. 33) 
be the original direc- 
tion of projection and 
Vo the initial velocity. 
Call the angle of ele- 
vation RO X, 6. 
Let x,y, be the co- 
F^<^- 22- ordinates of any point 

- P on the curve and / 

the time taken by the projectile to reach that point. Then 
(i) X = (Vo cos d) t 
(2) y ^ (Fo sin 0) t - ^gP. 
By eliminating / we have 

g 




(3) y = X tan — 



x^ which is the equation 



2l'V cos^ 6 
of the trajectory. 

This is the equation of a parabola with a vertical axis. 

From (i), (2), (3) can be found the value of the horizontal and vertical 
ranges, the time of flight and the elevation for maximum horizontal range. 

134. Effects of Air Resistance. —The resistance of the air causes 
a large deviation from the parabolic trajectory and a great diminution of 
range in the case of rapidly moving projectiles. The maximum horizontal 
range is theoretically obtained when the angle of elevation is 45°, but this is 
practically true only for low velocities. For swift projectiles this angle is 
about 35°. 

Other effects of air-pressure on a projectile are (a) gyroscopic deviation 
(drift) with elongated projectile from rifled gun, since the pressure tends 
to tilt the top of the projectile upward; (b) lateral deviation from the plane 
of projection in case of a ball rotating about a vertical axis, caused l)\ un- 
equal air-pressure on opposite vertical halves according as the velocity 
at the surface due to rotation is with or against the motion of translation; 
as in the case of curve pitching in baseball. 

In the case of an unsymmetrical rotating ])rojectik- tlie air-resistance 
may give rise to a very complex path, as, for exanipic, in the Australian 
boomerang, a curved club which returns to the place from whic h il is thrown. 

135- Velocity acquired in descending Frictionless Inclined 

/ H 

Plane. — Since v =\ 2i7s , if -^ == /^, that is, if tlu^ hodv traverses the 

whole length of the plane, V = V2gll . 'V\\\> is indeiuMulenl of />, the 



PENDULUM, 63 

length of the plane, and is equal to the velocity acquired by a freely falling 
body in descending through the vertical height H. 

136. General Proposition. — A body descending from a given point 
to a given horizontal plane will acquire the same velocity whether this descent 
is made vertically or ohliqnely over an inclined plane or over a curved surface. 
This proposition is, of course, departed from in practice, because of the 
effect of friction and other resistances. 

137. Properties of Cycloid. —The path along which a body will 
descend most swiftly, between two points not in the same vertical, is an 
inverted cycloid passing through these points with its cusp at the upper- 
most of them. For this reason, the cycloid is often called the Br achy s- 
tochrone. 

Another important property of the cycloid is, that the time required to 
descend to the lowest point of the complete inverted curve is the same, from 
whatever point of the curve the body may start. The cycloid is therefore 
a Taiitochrone. 



PENDULUM. 

138. Case of Body rolling on Curve or Suspended by Flexible 
Cord.— See Figs. 34, 35- 

139. Simple or Mathemat- 
ical Pendulum This may be 

defined as a gravitating particle 
suspended by a cord without 
weight. 

140. Time of Vibration. — 
For very small circular arcs 




Fig. 35. 



t = 



W^. (^ 



— . (See§ 112, p. 51.) 



This is independent of amplitude. (GaKleo, about 1583.) 
For all circular arcs, 



h \ 3 
^/ 

a very close approximation used in practice. In the formula h is the 
versed sine of the arc of vibration. 

141. Isochronism. — If the particle moves in a cycloidal arc, the 

formula/ =~y/ — is true for all amphtudes. (Huygens, 1673.) (See 

o 

?S 113. P- 5I) for proof.) 



64 



MECHANICS. 




Hence, for a cycloidal pendulum, or approximately for a circular pen- 
dulum vibrating through 
small arcs, the vibrations 
are isochronous; that is, 
performed in equal times, 
independently of the am- 
pHtude. 

Construction of cycloidal 
pendulum. (See Fig. 36.) 

142. Laws of Peiidu= 
lum. — 

I. Time is independent 
of amplitude (subject to 
limitations stated above). 
II. Time varies as the square root of length. 

III. Time varies inversely as the square root of g. 

IV. Time is independent of mass or material, of pendulum. 

143. Physical Pendulum.— The time of vibration is varied by changes 
in the distribution of the mass of the pendulum. 

144. Center of Oscillation.— This is that point of a physical 
pendulum which vibrates in the same time as if it were free from all con- 
nection with the remaining particles. Its position can be determined 
mathematically for homogeneous bodies of regular form. In a prismatic 
rod suspended at one end, it is at a distance of two-thirds the length of the 
rod from the point of suspension. The length of a physical pendulum is 
the length of the equivalent simple pendulum, and is equal to the distance 
from the axis of suspension to the center of oscillation. 

It can be shown that for any physical pendulum this is equal to the 
moment of inertia of the system relatively to the axis of suspension, divided 
by the product of the mass of the system into the distance from the axis of 
suspension to the center of gravity. (See § i45-) 

The center of oscillation can also be shown to be the center of percussion, 
which is the point at which a body suspended from an axis may be struck 
a blow in its plane of rotation without producing any pressure upon the 
axis. 

The center of oscillation and axis of suspension are mutually convertible; 
that is, the time of oscillation is the same from whichever of these points 
the pendulum is suspended. (See § 146, p. 65, for demonstration.) 

These properties of the physical pendulum were discovered by Huygens 
and announced in 1673. 

145. Length of Equivalent Simple Pendulum. — The length of 
the simple pendulum whose time of vibration is the same as that of a par- 
ticular physical pendulum may be found as follows: — 



PROPERTIES OF PHYSICAL PENDULUM. 



65 



Call M the mass of the given physical pendulum, 
and Ip its moment of inertia relative to the point of 
suspension P (Fig. 37). Let G be its center of 
gravity and O its center of oscillation when sus- 
pended from P. The distance PO is the required 
length, /. With the equivalent simple pendulum 
the mass M will be concentrated at O. Let PG 
= d. 

It is necessary and sufficient for equality in the 
periods of the physical and equivalent simple 
pendulum that for every value of the angle of de- 
flection 6 the angular acceleration a due to the 
torque produced by the weight of the pendulum 
shall be the same for both. 




Fig. 37« 



dco 
~dt 



In general (§ 89, p. 38) Tt = loj whence Tdt = Idw, and T ^ I 

= Ia. 

For the physical pendulum the moment of the weight, acting through 

G at the instant when the deflection is 6, is Tp = Mgd sin d ^ Ip ap, and 

for the simple pendulum, 7^^= Mgl sin d = h as^ MP as, 

, Mgd sin d Mgl sin ^ ^ ^. . • ^ r . 

whence ap = — ^—^ , as = — ^^tTo ^^^ equality of period of vi- 



MP 



bration ap = as, whence / = 



h 



Md 



The following examples will illustrate the application of the preceding 
principles : — 

1. Position of Center of Oscillation of Prismatic Rod of length L, sus- 
pended from one end. 

/, = ^MD. Ip-^I, + M {hiy =iMD',l^ j^^ 

2. Position of Center of Oscillation of Sphere suspended by very fine 
wire of length h. 

Ip , . r' 



\MD 
M\L 



3 ^• 



%Mr\ 



Ip = Ig+M{h +ry;l = 



M{h + r) 



^ h + r -\- 



h + r 



146. Convertibility of Point of Suspen- 
sion and Center of Oscillation. — That the 
time of vibration is the same from whichever 
point, P or O (Fig. 38) the vibrating mass is 
suspended follows from the fact that the length 
of the equivalent simple pendulum is the same 
in either case. Denote by kg, kp, ko, the radii of 
gyration of the mass relatively to G, P and O 
respectively. The length of the equivalent simple 
pendulum when P is the point of suspension is 




Fig. 38. 



/* = 



h 



Mki 



Mkg^ + Md^ kg^ , , , 



Md 



d 



Md Aid 

When O is the point of suspension 

/,, Mko^ Mkg^+Mdi" kg^ ^ ^ ^ 

=^ + d, (2). 



/. -= 



Mdj^ Md^ 



Md, 



k ■ 
Sinre L — d 4- d^ we have from (j) kJ^ = dd,. Henre L = ~ '^- 



^ fl = 



66 



MECHANICS. 






+ 6?i = d -]- dj = Ip; that is, whether the vibrating mass is suspended 




AI' 



Fig. 39. 



from P or from O the length of the equivalent simple pendulum is the same, 
and hence the time of vibration is the same. 

147. Metronome Pendulum. — A pendulum in which a consider- 

able portion of the mass is situated above the axis of 
suspension is called a metronome pendulum. By vary- 
ing the position of a movable mass i/' (Fig. 39), the 
time of oscillation and length of the equivalent simple 
pendulum can be varied within very wide limits, 
without increasing the dimensions of the apparatus. 
The metronome of Ma^lzel (181 6) commonly used for 
giving time in music, is constructed on this principle, 
and is obviously far more convenient than the ordinary 
"bullet" pendulum previously employed. 

It is evident that with this form of pendulum the 
center of oscillation lies outside of and below the mass of the pendulum it- 
self. Its position can be calculated by means of the formula in § 145, p. 65. 

148. Determination of Length of Pendulum beating Seconds. 

■ — I. Borda's Method. (1790.) 

Invariable Pendulum. — Ball of 
platinum B (Fig. 41) is sus- 
pended by wire so light that it 
is without sensible influence on 
time of vibration. Length of 
wire {h) and radius of ball (r) 
are measured. 

Distance of center of oscilla- 
tion O below center of sphere is 
given by analysis. It is x = 

2 r^ 
— T ; the length of the equiva- 
lent simple pendulum is therefore 

2 r^ 
I = h +r + - T— -• 
5 ^+^ 
The time / is determined by 

comparison with a clock. Then, 
caUing / the distance, AO, Is the 
length of the seconds pendulum, 
^:i : 'VT: VTs. 

The time / is determined by 
the method of coincidences. 

Fig. 42 shows certain details 
of the apparatus. 

11. Kater's Method. (1818.) 

. /.'y»r ii-)i'i- i*/i//-> J^/?'H/i III II III T*Vi<a 



O A 



\^B 



E' 




PENDULUM MEASUREMENTS. 6'/ 

positions of the weights A, B, (Fig. 40) are varied until the pendulum 
vibrates in the same time whether hung upon E or E\ Hence, when E 
is the axis of suspension, E^ is the corresponding center of oscillation. 
The length of the equivalent simple pendulum is, therefore, the distance 
between E and E\ The time of vibration is measured by a clock, and 
the length of the seconds pendulum determined as in I. 

With a reversible pendulum, if the time of vibration is approximately 
but not exactly, the same when suspended from either knife-edge, which is 
in reality the usual case in practice, it can be shown that the length of the 
equivalent simple pendulum is 

c — J 

(tit J 6^2^2 

when /i, 4 are the times of vibration, and d^, ^2 the distances of the center 
of gravity from the two knife-edges. 

Bessel (1826) used a modification of Borda's method. In this the length 
of the wire suspension is varied from d to d', and the corre- 
sponding times of vibration t, t' are determined. 

Then I - V = {d - d') (1 - ~^J, and vT: VF : : t : V. 

Hence we can determine / and V. d, d^ = distances from axis 
of suspension to center of sphere. The practical advantage is 
that d—d' is the principal term, a quantity more easily measured 
than either d or d'. (See Fig. 43, in which AA' =^ d — d\) 

149. Corrections. — Corrections are applied to observed re- 
sults in pendulum measurements for: — 

1. Amphtude of arc of vibration. 

Semi-arc used by Kater (seconds pendulum) = i°i5''; by Fig. 43- 
Mendenhall (half-seconds pendulum) = 30'. For seconds 
pendulum, amplitude 48', correction is 1.05 seconds per day. 

2. Temperatures of pendulum and of measuring rods. Lengths must 
be reduced to standard temperature. 

Use of "dummy" pendulum for former. "Pendulum coefficient" 
determined experimentally by swinging at known temperatures. 

3. Reduction to vacuo: corrections for buoyancy of air, air-drag, vis- 
cosity. 

Bessel devised a reversible pendulum, symmetrical in shape about its 
middle point and with the knife-edges equidistant therefrom. One end 
of the pendulum was made heavier than the other, and the knife-edges were 
placed so as to be approximately at corresponding centers of oscillation and 
suspension. With such a form the air-correction disappears in the calcu- 



--^' 



6 



68 



MECHANICS. 



lation. Fig. 44, p. 69, shows Repsold's Bessel's pendulum, Fig. 45 shows 
Peirce's (U. S. C. S.), which is hollow, but loaded at one end. 

In many cases it is preferable to swing the pendulum in an air-tight 
case under diminished pressure; e. g., 60 mm. at 0° C, as nearly as may 
be, and to reduce to this exactly, the pressure correction being determined 
experimentally for the individual pendulum. 

4. Clock correction. 

5. Flexibihty of support of pendulum. 

6. Reduction to sea level; as g diminishes with altitude. 

There are also certain other minor corrections which are necessary to 
ensure the greatest accuracy. 

150. Results of Measurement.— The following are measured values 
of the length of the pendulum beating seconds and of g at certain places, 
according to data given in the Reports of the Coast and Geodetic Survey of 
the United States. 



station 




Latitude t.\ 


evation 
m. 


Length 

cm. 


g 

en, 


I. 


Washington, U. S. C. 


& G. S. 


38°53'i3'' 


14 


99-3047 


980.098 


Boston, State House 




42°2i'33'' 


22 


99-3335 


980 


.382 


Paris, Observatory 




48°5o'i4'' 


74 


99-3863 


980, 


•903 


Greenwich, Observatory 


5i°28'4o'' 


48 


99-4134 


981, 


.171 


Berlin, Observatory 




52°3o'i6'' 


35 


99.4230 


981, 


.265 



/ 



151. Uses of Pendulum. — I. Determination of Value of g. 
(Huygens, Paris, 1673.) 

T TT^/ 

g ^ 

a. Determine U at different stations, (i) By direct measurement oi h 
(Huygens). (2) By comparison of period of invariable pendulum swung 

I 



n\/- 



at different places ; g 



(X 



/2 



(Bouguer, Andes, 1737.) 



Bouguer used a sphere suspended by a thread of constant length. Kater 
and Sabine (1820-25) employed an invariable seconds pendulum consisting 
of a bar terminated by a disc-shaped bob, the Indian Trigonometrical 
Survey (1865-75), a Repsold's Bessel inversion pendulum (Fig. 44). Von 
Sterneck (1880), and Mendenhall, U. S. C. & G. S. (1890) used half-seconds 
pendulums. Fig. 46 shows that of Mendenhall. A quarter-seconds pen- 
dulum has also been used. 

b. Study of diminution of g with elevation. 

(i) Determination of reduction factor to sea-level. (Bouguer, Andes, 

I737-) • 

(2) Determination of mean density of earth (J). (Carlini, Alps, 1821). 



MEASUREMENT OF GRAVITATION. 



69 



Conversely, determination of mean density of mountain from J. (Men- 
denhall, Fujiyama, 1880.) 

Mendenhall foimd density of Fuji = 2.02. Putnam (1894) found den- 
sity of Pike's Peak to be = 2.57 ; or J = 5.63, assuming d = 2.62. 



H 




W 



y 



Fig. 44. 



r\ 



M 



fl 



Fig. 45 



Fig. 46. 



70 MECHANICS. 

c. Study of change of g below surface of earth. (Airy, 1826, 1854 .) 
Frequency of seconds pendulum at depth of 1250 feet in Harton Coal 
Pit increased 2 J oscillations in 24 hours, denoting increase of ^ of i 
part in 19,200. 

From this it follows that g at surface is greater than if there were no 
shell of matter above base of pit by i part in 14,000; from which a value 
of A can be determined. 

Von Sterneck (1883) from observations down to a depth of 1,000 meters 
has determined a formula representing the variation of g with depth. 

The pendulum methods of determining A are not comparable in accuracy 
with the methods of Cavendish and Poynting. The same is true of the 
determination of A from observations on the deflection of the plumb line 
or spirit level by a mountain, — a method employed by Bouguer on Chim- 
borazo, in 1740, and Maskelyne on Schehallien, 1774. 

d. Study of local variations in g. Value is less than normal in mountain- 
ous regions and greater on islands and borders of continents. Observa- 
tions made among the Himalayas by Indian Survey up to an elevation of 
15,408 feet (Basevi). In Europe by Defforges, in United States by Defforges 
and Putnam, 

OBSERVATIONS OF PUTNAM. 







Elevation 


Observed value of g 

reduced to 

sea level 


Computed 
value of g 






■m. 


cm. 


cm. 


Washington 




14 


980.101 


980.087 


Denver 




1638 


979.941 


980.156 


Pike's Peak 




4293 


979.844 


980.083 


Yellowstone 


Canyon 


2386 


980.369 


980.605 



e. g is independent of mass and material. (Newton, 1687; Bessel, 
1832). Mass and material of bob of pendulurn varied; time of vibration 
remains unchanged. 

II. Determination of Figure of Earth. — (Richer, 1672; Newton, 
1687; Clairaut, 1743.) 

From the law of. variation of g with latitude it is possible to determine 

a — b 

the ellipticity or oblateness of the earth. Ellipticity s = . 

a 

Clairaut showed (1743) that for a spheroid of equilibrium of small oblate- 
ness composed of concentric strata each of the same density throughout, 
which, presumably, is approximately true of the earth as a whole, the 
ellipticity is £ = f ^^ — ^ where ^ is the ratio at the equator of the centri- 
fugal force to gravity and ^ the total fractional increase of gravity from 
equator to pole. At a much later date (1849) Stokes showed that for such 
a spheroid no particular law of density need be assumed. 



DETERMINATION OF FIGURE OF EARTH. 7 1 

Clairaut also showed that the value of g in any latitude X is given by the 
formula g^ = So ('^ + ^ sin^ X). It is easily seen that in the formula 



^ 



<b90 



^o sin^ X go 

It follows from these facts that the value of the polar flattening can be 
determined from a comparison of pendulum measurements in different 
latitudes. 

Approximately calling y = ^g^ and ^ = ^^^ (its value as found ex- 
perimentally by the pendulum), e = ^^. The calculations leading to 
exact results are of course very complex. The most probable value of s 

as found by the pendulum method is ^^ (Helmert, 1901, 1907). The 

geodetic method gives ;^ (Clarke, 1878), but this method is considered to 
be less trustworthy than the pendulum method owing to the comparatively 
small portion of the earth's surface which has been accurately triangulated. 
A very recent determination (1906) by the U. S. C. &G. S., using the geo- 
detic method and based on measurements in this country gives e = ;^" 
From a consideration of results of all the various methods which have been 

employed the value ^^ has recently been reached by Helmert (1907) 
as the most probable one, agreeing with the results from pendulum measure- 
ments alone. 

The difference between the greatest and the least value of g is about -^ 
of the minimum value, as stated above. 

The actual solid dealt with in all these calculations is the geoid, i. e.^ the 
spheroid whose surface would everywhere coincide with the ocean level. 

Formula. — The normal value of g for any latitude X, at sea-level, may 
be computed from the following formula given by Putnam (1897): — 

^X = 978.066 (i -f- 0.005243 sin^ X). 
The following particular values are thus obtained: — 

Place 



Washington, U. S. C. & G. S. 
Boston, State House 
Paris, Observatory 
Greenwich, Observatory 
Berlin, Observatory 

Values of g determined by actual measurement at the five particular 
stations given above are stated in § 150, p. 68. 



Latitude 


Value of g 




cm. 


0° 


978.066 


45° 


980.630 


90° 


983.194 


38°53'i3'' 


980.087 


42°2i'53" 


980.394 


48°5o'i4'' 


980.972 


5i°28'4o'' 


981.205 


52°3o'i6" 


981.294 



72 MECHANICS. 

These results should be reduced to sea-level for purposes of exact com- 
parison with the values calculated from the general formula. 

A later and perhaps somewhat more precise formula than the above is 
given by Helmert (1901); viz., 

g^ = 978.046 (i + 0.005302 sin^/^ — 0.000007 si^^ 2y^). 

The difference, however, in the results given by the two formulae is very 
slight. 

2h 

For an elevation h we may use the formula g^ = g (i ), or such 

R 

other as may seem preferable in any particular case. 

A formula which has been widely used for the reduction to sea-level is 

2h 
that of Bouguer, in which for the term of correction — is substituted 

R 

2h 3^ 

— (i 7). J is the mean densitv of the earth, and the densitv of 

R 4J 

the adjacent matter lying above sea-level. Strictly considered, Bouguer's 
correction applies to an extended plateau. The added term is introduced 
in order to take account of the attraction of the mass of elevated land on 
which the station is situated. 



VALUE OF g IN DIFFERENT LATITUDES. 
(From Everett's "C. G. S. System.") 



Spitzbergen 

St. Petersburg 

Berhn 

London 

Dunkirk 

Paris 

Padua 

Fiume 

Bordeaux 

Boston 

Washington 

Cape Town 

Port Jackson 

Calcutta 

Bombay 

Madras 

Sierra Leone 

Para 



Lat. 


£■. (measured) 


Elevation 




cm. 


w. 


79°5o' 


983.08 


6 


59°56' 


981.90 


8 


52°3i' 


981.27 


35 


5i°3i' 


981.18 


28 


5I°2' 


981.14 





48°5o' 


980.92 


70 


45°24' 


980.67 


31 


45°i9' 


980.63 


65 


44°5o' 


980.54 


17 


42°22' 


980.38 


22 


38°53' 


980.10 


10 


33°56'S 


• 97964 


10 


33°52'S 


979.67 




22°33' 


978.78 


6 


i8°54' 


978.61 


II 


i3°4' 


978.20 


8 


8°29' 


978.18 


58 


i°27'S 


978.03 


12 



regulation of clocks. 73 

Oceanic Islands. 

Ascension 
St. Helena 
Mauritius 

The values of g for London, Boston and Washington have been added 
to Everett's table from other sources. 

The following values of g were obtained by Putnam in connection with 
the M. I. T. party on the Peary Expedition to Greenland in 1896. A 
Mendenhall half-seconds pendulum was used. 



7°56'S 


978.29 


5 


i5°56'S 


978.65 


9 


20°I0'S 


978.87 







Latitude 


g ] 


Elevation 






cm. 


nt. 


Washington, U. S. C. & G. S. 


38°53'i3'' 


980.098 


14 


Sydney, C. B. 


46°o8'32'' 


980.720 


II 


Ashe Inlet, Hudson Strait 


6o°32'48'' 


982.105 


15 


Umanak, Greenland 


7o°4o'29'' 


982.590 


10 


Niantilik, Cumberland Sound 


64°53'3o'' 


980.273 


7 



III. Regulation of Clocks. 

The application of the pendulum to the regulation of clocks is generally 
accredited to Huygens (1656), although he appears to have been, in fact, 
anticipated by GaKleo. Rival claims are advanced for Hooke, Harris 
and others. ^ 

A circular pendulum with a spring suspension is universally used for 
this purpose. 

Escapement: Crown-wheel (De Vick, 1360, adopted for pendulum by 
Huygens, 1656); Anchor (Hooke, 1656); Dead Beat (Graham, 171 5); 
Gravity (Mudge, 1760); Free escapement, impulse communicated through 
suspension springs (Riefler, 1889). 

Compensation pendulums: Graham, mercurial (17 21); Harrison, 
gridiron, brass and iron rods (1726); Reid, modification using zinc and steel 
rods (181 2); Riefler, rod of invar (Guillaume's nickel-steel alloy) with 
short compensation tube of steel and brass (1898). 

A standard clock should be kept in a constant-temperature room. 

Compensation may be secured for barometric changes by the action of 
a magnet governed by a barometer upon a second magnet carried by the 
pendulum, as in the standard Greenwich Observatory clock. Or, better, 
the clock may be kept in an air-tight case under reduced pressure {e. g., 
675 mm.), and so protected from atmospheric variations. This was done 
as early as 1867 in the Berlin Observatory, and is now common. Such a 
clock is automatically wound by electricity. 

A standard astronomical clock by Riefler at the U. S. Naval Observatory, 
Washington, placed in an air-tight case has run with a mean daily variation 
of only 0.015 seconds for a period of 3 J months. This clock possessed a 
nickel-steel pendulum rod and was kept in a constant temperature room. 



74 MECHANICS. 



IV. Former Standard of British Weights and Measures. 

As the result of measurements in 1817 and subsequent years Kater had 
determined the length of the seconds pendulum at London reduced to 
vacuum and sea-level to be 39.1393 in. The British Weights and Measures 
Act of 1824 constituted as the legal standard of length for Great Britain 
the brass yard made by Bird in 1760, a Hne standard, in terms of which 
Kater 's pendulum had been measured. It furthermore provided that in 
case of loss or destruction of the standard yard it should be reproduced by 
constructing a new standard bearing the same proportion to the length of 
the seconds pendulum as that borne to it by the original standard. In 
1834 the Imperial standards of weights and measures, including the yard 
of 1760, were destroyed in consequence of the burning of the Parliament 
Houses. Meanwhile various sources of error had been discovered in Kater's 
pendulum measurements, and it became evident that a much closer approx- 
imation to the lost standard yard could be made from a comparison of the 
best existing copies than by employing the method of restoration set forth 
in the Act of 1824. A new Kne-standard bar of bronze (Baily's metal) was 
therefore constructed by the method of copying. This was constituted 
the legal Imperial standard yard by the Weights and Measures Act of 1855. 

Under the Act of 1824 the pound (Troy of 5760 grains) was defined by 
reference to the weight of a cubic inch of water at 62° F. as determined by 
Shuckburgh. 

The Imperial pound of 1855, an avoirdupois pound of 7000 grains is 
represented by a cylindrical mass of platinum, and is in no way referred 
to any other quantity. 

The length of the seconds pendulum, therefore, formed the legal basis 
of the British system of weights and measures only from 1824 to 1855. 

V. Physical Demonstration of the Rotation of the Earth. 
(Foucault, 1851.) 

If a pendulum could be suspended at the north pole directly in the line 
of the earth's axis, and set into vibration in any chosen plane, a line marking 
the horizontal trace of the plane of vibration in its original position would 
rotate with the earth in a left-handed direction at the rate of 15° per hour. 
Because of the permanence of the plane of vibration of the pendulum this 
would remain unchanged in position. Hence referred to the surface of 
the rotating earth the plane of vibration would appear to rotate right- 
handedly at the same rate. 



ROTATION OF EARTH. 



75 




In a lower latitude, as e. ^., at B (Fig. 47)^ the pendulum in the actual 
experiment is set into vibration in the meridian BN. Somewhat later B 
will have moved to B\ But the horizontal trace 
of the vertical plane in which the pendulum con- 
tinues to move remains parallel to its original 
direction, and is B'E, which is no longer north 
and south in direction, but inclined to the meridian 
by an angle NB'E dependent on the latitude. The 
hourly apparent rotation of the plane of vibration 
for latitude /I is ^ = 15° sin X. 

For X = 45°, d = 10° 36'. 

At the equator = o. 

At the poles (9 = 15°. ' 

A gyroscope suspended so as to move freely 
about a vertical axis may be used for the same 

purpose. The wheel with its axis horizontal is caused to rotate with great 
rapidity. If the axis is placed initially in the meridian, the revolution of 
the earth causes an apparent displacement of the plane of rotation of the 
wheel which is of the same character as that of the plane of vibration of the 
pendulum previously described. 

In Foucault's original experiment, performed in the Pantheon, Paris, 
the pendulum was 220 ft. in length. The motion was shown by causing 
a pointed spindle projecting below the cannon ball which formed the bob 
to swing through an arc of moist sand. 

Permanent records were obtained at this Institute in 1876 by the use of 
a smoked glass plate on which the trace of the plane of vibration was regis- 
tered by means of a style. 



Fig. 47. 



UG SO 1908 



NOTES 



ON 



MECHANICS 



BY 



CHARLES R. CROSS 



Printed for the use of Students 



IN THE 



Massachusetts Institute of Technology 



1908 



BOSTON 

WM. B. LIBBY, THE GARDEN PRESS 

16 ARLINGTON STREET 



I 



i 



